2013年11月23日 星期六

毀滅日本的人道原子彈

毀滅日本的人道原子彈

曾慶潭2013-11-24於紐西蘭
Ching-Tang Tseng
ilikeforth@gmail.com



這是眉口秋明(Mike Cuming)先生於2013-04-03所傳贈之大作的半篇譯文


What a piece of history Tinian Island, Pacific Ocean.. 
It's a small island, less than 40 square miles, a flat green dot in the vastness of Pacific blue.

首先,讓我們來回顧太平洋提尼安島的一小段歷史:
這是一個很小的島,面積只有不到40平方英哩,在浩瀚的藍太平洋中,它只是一個平坦的小綠圓點。(照片一)



Fly over it and you notice a slash across its north end of uninhabited bush, a long thin line that looks like an overgrown dirt runway. If you didn't know what it was, you wouldn't give it a second glance out your airplane window.

搭飛機飛越提尼安島時,你會注意到島上有一條斜線,線的北端消失在無人居住的灌木叢林中,它是一條很長的細線,看起來像是一條雜草叢生的髒亂跑道。如果你不知道它的相關歷史,你絕不會想再透過飛機窗口正眼瞧它第二眼。(照片二)(照片三)




On the ground, you see the runway isn't dirt but tarmac and crushed limestone, abandoned with weeds sticking out of it. Yet this is arguably the most historical airstrip on earth. This is where World War II was won. This is Runway Able:

從地面上仔細端詳,這條跑道看起來還不太髒亂,停機坪和碎水泥區則長出了雜草。然而,它可以算是地球上最具有歷史意義的一條飛機跑道。這條跑道曾經贏得第二次世界大戰,它被尊稱為亞伯跑道(Runway Able)(照片四)


On July 24, 1944, 30,000 US Marines landed on the beaches of Tinian .... Eight days later, over 8,000 of the 8,800 Japanese soldiers on the island were dead (vs. 328 Marines), and four months later the Seabees had built the busiest airfield of WWII - dubbed North Field - enabling B-29 Super fortresses to launch air attacks on the Philippines, Okinawa, and mainland Japan.

1944724日,3萬名美國海軍陸戰隊隊員登上提尼安島的海灘......八天之後,痴守島上的8800名日本士兵中,超過8000人在島上(與328海軍陸戰隊交戰)喪命。4個月後,美國海事工兵在此建成了這個二戰期間最繁忙的機場,它被尊稱為北機場(North Field),能讓B-29超級堡壘型轟炸機,直接對菲律賓、琉球和日本本土發動空中攻擊。

Late in the afternoon of August 5, 1945, a B-29 was maneuvered over a bomb loading pit, then after lengthy preparations, taxied to the east end of North Field's main runway, Runway Able, and at 2:45am in the early morning darkness of August 6, took off.

194585日午後,一架B-29,於炸彈裝載坑進行了裝載,經過很長的時間,才完成一切備便手續。然後,滑行到北機場主跑道──亞伯跑道的東端,並於86日上午245分,在清晨暗夜中起飛。

The B-29 was piloted by Col. Paul Tibbets of the US Army Air Force, who had named the plane after his mother, Enola Gay. The crew named the bomb they were carrying Little Boy. 6- hours later at 8:15am Japan time, the first atomic bomb was dropped on Hiroshima .

這架B-29的正駕駛是美國陸軍航空隊的上校保羅‧蒂貝次(Col. Paul Tibbets),為了紀念母親,他以母親之名伊諾拉‧蓋伊(Enola Gay)為這架飛機命名。航員則將他們這次裝載的特別炸彈命名為小男孩(Little Boy)。飛行了6小時之後,於日本當地時間8:15,他們在廣島(Hiroshima)投下了第一顆原子彈。

Three days later, in the pre-dawn hours of August 9, a B-29 named Bockscar (a pun on "boxcar" after its flight commander Capt. Fred Bock), piloted by Major Charles Sweeney took off from Runway Able. Finding its primary target of Kokura obscured by clouds, Sweeney proceeded to the secondary target of Nagasaki, over which, at 11:01am, bombardier Kermit Beahan released the atomic bomb dubbed Fat Man.

三天後,89日黎明前,另一架名為Bockscar(它是為了紀念飛行指揮官弗雷德·博克(Fred Bock)上尉而採用與棚車(boxcar)發音近似之雙關語所命名的),由查爾斯·斯威尼(Charles Sweeney)少校駕駛的B-29,從亞伯跑道起飛。原本要尋找第一個主要轟炸目標小倉(Kokura),但此目標被雲層遮住而看不清楚,於是改為前往第二個轟炸目標長崎(Nagasaki)。上午1101分,投彈手克米特·比漢(Kermit Beahan)投下了被稱為胖子(Fat Man)的原子彈。

Here is "Atomic Bomb Pit #1" where Little Boy was loaded onto Enola Gay:

照片五是『一號原子彈儲存坑』,『小男孩』從此坑加載到埃諾拉‧蓋伊B-29轟炸機上:


There are pictures displayed in the pit, now glass-enclosed. This one shows Little Boy being hoisted into Enola Gay's bomb bay.

此處有幾張照片顯示,原用來儲存原子彈的紀念坑,已使用玻璃罩蓋蓋於坑上。照片六顯示『小男孩』正被吊掛到埃諾拉‧蓋伊B-29轟炸機的炸彈艙。


And here on the other side of ramp is "Atomic Bomb Pit #2" where Fat Man was loaded onto Bockscar.

照片七顯示,另一邊的斜坡上為『2號原子彈儲存坑』。


照片八顯示『胖子』被加載到Bockscar B-29轟炸機上。


The commemorative plaque records that 16 hours after the nuking of Nagasaki , "On August 10, 1945 at 0300, the Japanese Emperor without his cabinet's consent decided to end the Pacific War."

以原子彈轟炸摧毀長崎之後16小時,根據日本紀念牌匾的記錄顯示:『194581003:00,日本天皇未經他的內閣同意,決定結束太平洋戰爭。』

Take a good look at these pictures, folks. This is where World War II ended with total victory of America over Japan . I was there all alone. There were no other visitors and no one lives anywhere near for miles. Visiting the Bomb Pits, walking along deserted Runway Able in solitude, was a moment of extraordinarily powerful solemnity.

各位,請好好看看這些照片。這是美國全面戰勝日本,結束了第二次世界大戰的地方。我獨自一人拜訪此地,沒有其他遊客,幾英哩內沒有人居住。光臨原子彈儲存坑,沿著廢棄的亞伯跑道獨行,那是個非常莊嚴的時刻。

It was a moment of deep reflection. Most people, when they think of Hiroshima and Nagasaki , reflect on the numbers of lives killed in the nuclear blasts - at least 70,000 and 50,000 respectively. Being here caused me to reflect on the number of lives saved - how many more Japanese and Americans would have died in a continuation of the war had the nukes not been dropped.

那是一個應該深刻反省的莊嚴時刻。大多數人,每當他們想起廣島和長崎時,總是只會反應在核爆炸中喪生了的生命數字──至少分別為70,00050,000人。在這裡,卻引起了我的另種反思,以原子彈轟炸日本,確實證明了反而是挽救了更多人之性命的核爆。否則,日本人和美國人的戰爭若再延續,而沒有從此處開飛機去投下原子彈,雙方還得犧牲多少人的性命?


以下為本人的補述:

『大東南亞共同受害圈』在二次大戰期間,被日本人屠殺了五千九百多萬人( >59,000,000),而且大部份都是無辜的平民老百姓。日本戰敗投降後,對『大東南亞共同受害圈』內各國,至今分文未賠。二戰之後,如果要讓真正的民主,在整個世界普及起來,就不應該由任何個人,包括蔣中正在內,草率決定處置日本人的方式。現在,『大東南亞共同受害圈』內各國,有權以現代民主的態度,向日本討回五千九百萬死難同胞的血債。

眉口秋明(Mike Cumming)先生最近傳送給我的這篇文章,為什麼我只翻譯了半篇?

後半篇,他以塞班島上的自殺崖(Suicide Cliff)史實,證明日本人喪盡天良、毫無人性。因此,加強闡述了『非用原子彈來消滅日本不可』的人道依據。『大東南亞共同受害圈』內的受害國家,並不需要另找史實來證明日本鬼子的惡德惡行,所以我就不再多譯後半篇文章。

相對於日本的德國,他們賠過一次大戰後的戰敗賠款,賠了92年,直到2010年才還清所有的天文數字債務,不管怎麼賠,德國人畢竟是以真正符合民主要求的態度賠淨了賠款。大家反過來看看日本,日本鬼子算是什麼東西?

現在,日本人還不顧別人死活,硬要在島的東邊──必定會發生大海嘯的區域,蓋了一大堆原子爐,出事,就將半衰期接近兩千年的高強核輻射物質往太平洋排放,台灣人首先永久受害。

我在參與中華民國核能技術發展工作期間,明確的知道,全體台灣人講求世界道德,絕不考慮在台灣東海岸設置任何原子爐。將心比心,以台灣人的心比日本人的心,以德國人的心比日本人的心,大家就能明白,日本人惡毒到了什麼程度?

不久之前,日本人向英國及法國大批買進能直接製造原子彈的鈽原料15噸。關於鈽與原子彈間的關係,明確的理論數據是:5.8公斤鈽,就能製造一枚原子彈。

但是,英、法也不是傻瓜,賣給日本的鈽,並不是什麼好東西,因為,裡面含有大量高於可容忍度的另種鈽的骯髒同位素,我們稱其為中子毒物,它只吃中子卻不貢獻於核分裂,而且影響程度還特別高。因為是同位素,『骯髒鈽』就難以與能造原子彈的『標的鈽』分離開來。如果強行使用這種髒鈽來製造原子彈,造出來的就會是威力極差的『髒彈』。使用鈾為燃料的核能電廠,運轉初期,好的標的鈽會逐漸增加,鈾燃耗到某一限值之後,標的鈽的產量反將下降,前述骯髒鈽的產量則只增不減,核能電廠中的鈾,燃耗得越久,鈽的品質就會越差。這也就是英、法兩國,會將大量無用而難再進行後續處理的劣質鈽,賣給日本的原因。

日本人將髒鈽買到手後,卻惡毒起來,硬將這種骯髒鈽送進超過35年高齡、早該退休的臨終原子爐內使用。現在,出事了,竟不斷地將難以想像的髒鈽排進太平洋,並堅拒有能力分析此狀況的人深究日本鬼子的責任。

每年到了311,大家必定又要紀念大海嘯了,全面落定確認311就是『天譴日本』,絕對正確。天要譴死這個喪盡天良的日本鬼子時,誰能阻擋?如果日本鬼子繼續在東海岸設置原子爐,那我每年都可以寫一篇替天行道的文章同來紀念。

日本人如果惡劣到敢再發動戰爭,或者是惡劣到敢搞核子武器,那麼,我建議俄羅斯、北韓、中共,同時向日本直接投下足夠立即毀滅整個日本的原子彈,全都可以算是人道原子彈。

日本若再堅不撤除東海岸的原子爐,那麼,上述各國也可以開始考慮,就派潛艇拖送原子彈,到日本東邊的太平洋內,直接試爆,再造大海嘯,可以造成如同『311天譴日本式』的大海嘯,只毀日本而別國少受損害,更可以算是人道試爆的原子彈。

這篇網文可以永留網上,專供患有癡呆症的日本人牢記史實,甚至於應該將這些史實內容,編進日本人的國民教科書中,永久提醒日本人,投在廣島與長崎的人道原子彈是怎麼來的?日本人,你搞清楚了沒有?

癡狂的日本人,你敢再造次?以後的原子彈,都更進步到不再需要提尼安島上的亞伯跑道了。美國投下的人道原子彈,是無法竄改的歷史,你不該讓世人覺得只能用人道原子彈來解決你的問題。你想找美國報仇?再搞日俄戰爭?再度侵略東南亞?門都沒有!學通中國人的道德與愛心吧!我才會反過來幫你們講好話。

大東南亞共同受害圈內的任何國家,除了該討戰爭債外,都有資格教你們如何治理國家,因為,大家都活得好好的,只有你們才活得真不像樣。


2013年2月12日 星期二

A High-Precision Floating-Point Computational System in Win32Forth

A High-Precision Floating-Point Computational System in Win32Forth
(Adjustable few hundred decimal digits)

Ching-Tang Tseng
2013/2/12 Hamilton, New Zealand


(1).  General description

2013/02/12
 
Once upon a time, I’ve got a high-precision floating point system. I typed

PRINT 1000*0.228

This was the answer:

227.9999999999999999757

Uh hoh….. What would be your reaction? To me, unsatisfied, it should be 228. This system was not good enough. If I want a system done right, I have to design it myself.

For more than 30 years, FORTH is the one and only one programming language I preferred to use. Most of the time, When I used FORTH, I was concentrated on mathematic calculation application. During the past 4 years, ABC FORTH has been built by me. This is a BASIC programming style system in FORTH, easy for using in math, and is designed by FORTH high level definitions only. I built it step by step, a lot of public domain beautiful FORTH codes, created by the FORTH sages and able men of the past, had been used in this system.  

4 years ago, I would like to do a non-profit research by FORTH. It is about the topic: To forecast the big earthquake a few days early. Signal analysis and math calculation with hardware device operation are needed in the system. And may be, to collect much enough data by automatic internet connection function should be included as well. But I have no money to buy any kinds of software or hardware. Win32Forth system and my personal technique are all I have, during these days.

After 4 years development, ABC FORTH performance is good enough for math using and for my personal research application. There were many articles related to ABC FORTH had been posted on this Blog. They were written in Chinese. All of them were going to let you know what I have done in the ABC FORTH?

There are six independent math systems in the ABC FORTH already. They are:

(1)   traditional integer number
(2)   traditional floating point number
(3)   complex number
(4)   fraction number
(5)   big integer number
(6)   big floating point number

When I developed my ABC FORTH, there was no supporting from anywhere or anybody. I have to do all testing by myself. Then, many practical application programs had been implemented out simultaneously. For the purpose of paying all my attention on the system development, I have no time and have no interesting to introduce all achievements in public domain.

Recently, a prototype of many digits high precision floating point arithmetic function has been carried out and has been added into my ABC FORTH. Now, I am able to use BASIC style program to deal with all math problems for high precision purpose. This is just a prototype system temporarily. I am going to improve it from now on. And wish it could be a nice tool for high precision math study in near future.

This is the first time to introduce ABC FORTH in English. Here, I would like to post some testing program for big floating point number computation and their output results instead of my poor English discussion. The following testing will be posted step by step but not in once.

Wish you enjoy it.

In this system, the precision of big floating point number is adjustable, independent with hardware math co-processor. A few hundred or thousand digits precision could be adjusted out by just to set the value of one variable named “CellsSetting” only. I am running Win32Forth in W7.

In this system, I am concerned about the accuracy only. As far as how much memory needed? How fast the running speed should be? Both of them are not concerned by me. When I set this system to be one hundred and twelve digits precision, I am able to get the output on the screen almost just after touch ENTER. Speeds are acceptable and response does make sense. Except for special condition, 112 digits precision will be a fixed setting in the program.       

What is an adjustable precision function? For example, you are able to get the value of sine and let his precision higher up to more than thousand digits. This is such a value of sin ( 1 ).

ok
BIGF1 BIGFSIN BF.
8.414709848078965066525023216302989996225630607983
71065672751709991910404391239668948639743543052695
85434903790792067429325911892099189888119341032772
92124094807919558267666069999077640119784087827325
66347484802870298656157017962455394893572924670127
08648628105338203056137721820386844966776167426623
90133827533979567642555654779639897648243286902756
96429120630058303651523031278255289853264851398193
45213597095596206217211481444178105760107567413664
80550089167266058041400780623930703718779562612888
04636081734524656391420252404187763420749206952007
71334780981427902145268255663208233521544160916442
09058929870224733844604489723713979912740819247250
48855487311931035068190815153260745739291118331962
82150897348688114214528382298651257016673840744551
92375614322129060592482739703681801585630905432667
84643107531263812173256701985601106836028901895019
42151616655191791451720046686595971691072197805885
40646001994013701405309580855205280525317113323054
61638363601816994797150048515079398383039567816794
81612214022089169871097439312119047662675566086294
39084
X10^ -1
 ok

According to the primitive function FSIN in the same Win32Forth system, we have

20 SIGDIGITS !  ok
1E0 FSIN FE. 841.47098480789644800E-3  ok

Directly manipulation above is just let the system seems to be the same as a calculator only. Here is a very fundamental programmable example. Let’s get the square root of the numbers from 1 to 10, and all of them, the precision are more than 100 digits.

Before I post the next testing material, these output results will be kept here.

And, I am sure, if I let my system to do this calculation: 1000 * 0.228. The output should be 228 exactly. But I would like to show it as 2.28 X10^ 2.

This is a very high precision floating point system.

2 BIGREALS AA BB

: T34
  11 0
  DO
     H{{ AA = I>BIGF ( I ) }}H
     H{{ BB = SQRT ( AA ) }}H
     CR ." SQRT( " I . ." )="
     BB BF.
  LOOP
  ;

 ok
T34
SQRT( 0 )=
0.0
X10^ 0

SQRT( 1 )=
1.0
X10^ 0

SQRT( 2 )=
1.414213562373095048801688724209698078569671875376
94807317667973799073247846210703885038753432764157
273501384623
X10^ 0

SQRT( 3 )=
1.732050807568877293527446341505872366942805253810
38062805580697945193301690880003708114618675724857
5675626141415
X10^ 0

SQRT( 4 )=
2.0
X10^ 0

SQRT( 5 )=
2.236067977499789696409173668731276235440618359611
52572427089724541052092563780489941441440837878227
496950817615
X10^ 0

SQRT( 6 )=
2.449489742783178098197284074705891391965947480656
67012843269256725096037745731502653985943310464023
4818594601226
X10^ 0

SQRT( 7 )=
2.645751311064590590501615753639260425710259183082
45018036833445920106882323028362776039288647454361
0615064578338
X10^ 0

SQRT( 8 )=
2.828427124746190097603377448419396157139343750753
89614635335947598146495692421407770077506865528314
5470027692461
X10^ 0

SQRT( 9 )=
3.0
X10^ 0

SQRT( 10 )=
3.162277660168379331998893544432718533719555139325
21682685750485279259443863923822134424810837930029
5187347284152
X10^ 0

Go on to the next section
\ *************************************************
To be continued.

(2). I/O format

2013/02/16

At the beginning, when I started to develop this system. I knew. The most difficulties of coding are dealing with the problem of how to define a few I/O instructions? They are helped to input and output the big floating point numbers into and out of the system. All of their function must work normally, no matter the system is under interpreted or compiled mode.

For output, I made my mind, let all big floating point numbers to be printed out with only one kind of format. This format has been showed above already. 50 characters per line are fixed. The decimal point of fraction part is always to be fixed at the position after the first mantissa integer number. Follow the mantissa, another line is used for ten to the power. It has been showed as is started with X10^ and ended with another minus or positive integer.

Two fundamental words for output are:

BF.           ( addr exp -- )        “Big Floating point number Dot”
addr means the beginning address of mantissa
exp means the exponent of big floating point number

BF@.       ( addr -- )               “Big Floating point number Fetch Dot”
addr means the beginning address of a big floating point number

As for input, it is more complicated than output. Lots of facts you have to consider. Lots of conditions you have to face to. Lots of frustration I have suffered while coding was progressed. Of course, at last, there were a lot of instructions I have finished. All of them are helped to convert many kinds of number’s format into a big floating point number, and then let my system is able to keep going to have successful operation on it.

For the time being, I would like to introduce only one kind of the input format to you. This format is good enough to let me easy to finish the discussion of this article and is apt to remember for the future using.

No matter how simple or how many or how big the input number is, this format is always to be represented as following:

S” 1.0”X10^ 1 POWER
S” 1.23” X10^ 0 POWER
S” -9.87654321” X10^ -397 POWER
S” 9.999999999999999999999999999999999999999999999” X10^ 0 POWER

The rules are:
(1)   Mantissa is always quoted by one pair of instructions S” ………”  
(2)   The format of mantissa is always in terms of the same format as output. The decimal point should be place at the position following the first digit. Leading negative mark is permit. Number is delimited by a double quotation mark, has been showed as above.
(3)   X10^ is one instruction. It means “times ten to the power of”.
(4)   Exponent is a general integer number.
(5)   All of all are ended with another instruction named POWER.
(6) If the input number is too big, too long or longer than one line width in one screen, this kind of number could be cut into many sections, other two instructions have to use. They are FIRST-SECTION and NEXT-SECTION.

A typical BASIC style simple program and its output result after execution can tell you how did this program to reach these rules?
In this program, “(“and “)”, both of them are unnecessary everywhere. They are using for the purpose of easy to identify there.

3 BIGREALS AA BB CC

: T40
  BASIC
10 LET H{ AA = ( S" 1.23" X10^ 1 POWER ) + ( S" 4.56" X10^ 1 POWER ) }H
   ::  H{ BB = ( S" 7.89" X10^ 5 POWER ) }H
   ::  H{ CC = AA / BB + AA }H
20 RUN CC BF.
30 LET H{ CC = NEGATE ( CC ) }H
40 RUN CC BF.
50 LET H{ CC = ABS ( CC ) }H
60 RUN CC BF.
70 END
;
 ok
T40
5.790007338403041825095057034220532319391634980988
59315589353612167300380228136882129277566539923954
3726235741444
X10^ 1

-5.79000733840304182509505703422053231939163498098
85931558935361216730038022813688212927756653992395
43726235741444
X10^ 1

5.790007338403041825095057034220532319391634980988
59315589353612167300380228136882129277566539923954
3726235741444
X10^ 1
 ok

As to the data structure, for the present, only BIGREALS has been used in this article.

Except BIGREALS,
BIGREAL, BIGFVARIABLE, BIGFVARIABLES, BIGFVALUE, BIGFVALUES, BIGFCONSTANT had been built as well.

No need to introduce you the others now.

Go on to the next section.
\ *******************************************************
To be continued

(3). Demonstration of Application

2013/2/18

\ Error function program:

\ MIMI: mini mini tiny register            for checking of lower bound
\ XXR : x Argument register                      for input
\ YYR : y = f(x) Accumulator                    for output
\ WKR : Working register                 for working in one term
\ NR1 : Numerator register 1
\ NR2 : Numerator register 2
\ NR3 : Numerator register 3
\ DR1 : Denominator register 1
\ DR2 : Denominator register 2
\ DR3 : Denominator register 3

10 BIGREALS MIMI XXR YYR WKR NR1 NR2 NR3 DR1 DR2 DR3

CellsSetting 10 * NEGATE VALUE TINY        \ to be fixed while loading

\ CONSTANT 10 was changed for ERFC as follows.
\ x=12 --> use 4, x=14 --> use 6, x=16 --> use 8, x=17 --> use 10

: InputData
  BASIC
10 LET H{ MIMI = S" 1.0" X10^ TINY POWER }H
20 LET H{ XXR = S" 3.0" X10^ 1 POWER }H    \ 17.0 ~~> huge #, OK for erfc
\ 30 PRINT " MIMI = :"
\ 40 RUN MIMI BF.
50 PRINT " X = :"
60 RUN XXR BF.
\ 70 PRINT " ERF(x) = :"
80 PRINT " ERFC(x) = :"
90 END
;

\ erf(x) = (2/sqrt(pi)) * ( integration(0-->x) of (exp(-t^2) dt ) )
\ = (2x/sqrt(pi)) * [ 1 - x^2/1!*3 + x^4/2!*5 - x^6/3!*7 + ..... ]

\ 0 <= x <= 11 OK for erf.
\ 4 <= x <= 11 OK for erfc, if needed, erfc(x)=1-erf(x)

: ERF
\  H{{ XXR }}H BIGF!                \ ( addr exp -- addr exp )
  BASIC
10 RUN InputData
20 LET H{ NR1 = BIGF1 }H
   ::  H{ DR1 = BIGF0 }H
   ::  H{ DR2 = BIGF1 }H
   ::  H{ NR2 = BIGF1 }H
   ::  H{ YYR = BIGF1  }H
30 LET H{ DR1 = DR1 + BIGF1 }H
   ::  H{ DR2 = DR2 * DR1 }H
   ::  H{ NR1 = NR1 * XXR * XXR }H
   ::  H{ NR2 = BIGF2 * DR1 + BIGF1 }H
   ::  H{ WKR = NR1 / ( DR2 * NR2 ) }H
   ::  H{ YYR = YYR - WKR }H
40 LET H{ DR1 = DR1 + BIGF1 }H
   ::  H{ DR2 = DR2 * DR1 }H
   ::  H{ NR1 = NR1 * XXR * XXR }H
   ::  H{ NR2 = BIGF2 * DR1 + BIGF1 }H
   ::  H{ WKR = NR1 / ( DR2 * NR2 ) }H
   ::  H{ YYR = YYR + WKR }H
50 IF  H{ WKR > MIMI }H THEN -30
60 LET H{ YYR = YYR * ( BIGF2 * XXR ) / SQRT ( PI ) }H
\ 65 let h{ yyr = bigf1 - yyr }h    \ for erfc(x<11)
70 RUN YYR BF.
80 END
\ YYR                       \ ( addr exp -- addr exp )
;

\ erfc(x) = [1/(exp(x*x)*sqrt(pi)*x)] * [ 1 - 1/2x^2 + 1*3/(2x^2)^2
\                                           - 1*3*5/(2x^2)^3 + ..... ]
\ x = 17 --> huge x, OK for erfc

: ERFC
  BASIC
10 RUN InputData
20 LET H{ NR1 = BIGF1 }H
   ::  H{ NR2 = BIGF1 }H
   ::  H{ DR1 = BIGF2 * XXR * XXR }H
   ::  H{ DR2 = DR1 }H
   ::  H{ YYR = BIGF1 - ( NR2 / DR2 ) }H
30 LET H{ DR2 = DR2 * DR1 }H
   ::  H{ NR1 = NR1 + BIGF2 }H
   ::  H{ NR2 = NR2 * NR1 }H
   ::  H{ WKR = NR2 / DR2 }H
   ::  H{ YYR = YYR + WKR }H
40 LET H{ DR2 = DR2 * DR1 }H
   ::  H{ NR1 = NR1 + BIGF2 }H
   ::  H{ NR2 = NR2 * NR1 }H
   ::  H{ WKR = NR2 / DR2 }H
   ::  H{ YYR = YYR - WKR }H
\ 45 RUN WKR BF.                    \ See divergent point from MIMI
50 IF  H{ WKR > MIMI }H THEN -30
60 LET H{ DR3 = EXP ( XXR * XXR ) }H
    :: H{ DR3 = DR3 * SQRT ( PI ) }H
    :: H{ DR3 = DR3 * XXR }H
70 LET H{ YYR = YYR / DR3 }H
80 RUN YYR BF.
90 END
;

\S
\ *********************************************************
\ Testing results:

erf
X = :
0.0
X10^ 0

ERF(x) = :
0.0
X10^ 0

erf
X = :
1.0
X10^ -1

ERF(x) = :
1.124629160182848922032750717439683832216962991597
02547534494144817599244025531359170221174136405196
9494684444739
X10^ -1

erf
X = :
2.0
X10^ -1

ERF(x) = :
2.227025892104784541401390068001438163882690384302
27605620935023888363674271912182870353723778740700
3625084786272
X10^ -1

erf
X = :
3.0
X10^ -1

ERF(x) = :
3.286267594591274276389140478667565511699180962626
75822609143139853710579890618085895322222697478706
2285426775476
X10^ -1

erf
X = :
4.0
X10^ -1

ERF(x) = :
4.283923550466684551036038453201724441218629285225
90383495086346113333368505671802422726158110538627
1012415050084
X10^ -1

erf
X = :
5.0
X10^ -1

ERF(x) = :
5.204998778130465376827466538919645287364515757579
63700058805725647193521716853570914788218734787757
0329661243901
X10^ -1

erf
X = :
1.0
X10^ 0

ERF(x) = :
8.427007929497148693412206350826092592960669979663
02908459937897834717254096010841261983325348144888
4541582615366
X10^ -1

erf
X = :
2.0
X10^ 0

ERF(x) = :
9.953222650189527341620692563672529286108917970400
60076738352326200437280719995177367629008019680680
487939328715
X10^ -1

erf
X = :
3.0
X10^ 0

ERF(x) = :
9.999779095030014145586272238704176796201522929126
00750342761045157057543316379867732183745349184683
712856553818
X10^ -1

erf
X = :
4.0
X10^ 0

ERF(x) = :
9.999999845827420997199811478403265131159514278547
46410808831657095005786958973188745927234986543819
418211441985
X10^ -1

erf
X = :
5.0
X10^ 0

ERF(x) = :
9.999999999984625402055719651498116565146166211098
81949685276620069312085944079608635441308535281856
140283421212
X10^ -1

erf
X = :
6.0
X10^ 0

ERF(x) = :
9.999999999999999784802632875010868834066496008126
15369522485938311457899472107948943662761515072149
437615991014
X10^ -1

erf
X = :
7.0
X10^ 0

ERF(x) = :
9.999999999999999999999581617439222058560138598977
61000677499703825861875398479583656469504534922860
172790264017
X10^ -1

erf
X = :
8.0
X10^ 0

ERF(x) = :
9.999999999999999999999999999887757028270170729200
32111556829720906568070835521036614058191866173481
9237285022
X10^ -1

erf
X = :
9.0
X10^ 0

ERF(x) = :
9.999999999999999999999999999999999995862968253486
18976194609653263747540428980389591497158310489836
00184378937
X10^ -1

erf
X = :
1.0
X10^ 1

ERF(x) = :
9.999999999999999999999999999999999999999999979115
12416237455242999213691611093205070508043921132553
652299156938
X10^ -1

erf
X = :
1.1
X10^ 1

ERF(x) = :
9.999999999999999999999999999999999999999999999999
99998559133974903630731655919342161305061590478873
27341811488
X10^ -1

erf
X = :
1.2
X10^ 1

ERF(x) = :          \ ***** Beware! this is a wrong result! *****
1.000000000000000000000000000000000000000000000000  00028964232747623645928111862273111180446503830761
9596656136387
X10^ 0

\ **************************************************
\ When 11<x<17, their precision are not the same as others.

erfc
MIMI = :
1.0
X10^ -48

X = :
1.2
X10^ 1

ERFC(x) = :
1.356261169205904212780306156590417572666782233287
92211903650655213650444305226488837167452692414329
9360066305547
X10^ -64

erfc
MIMI = :
1.0
X10^ -72

X = :
1.4
X10^ 1

ERFC(x) = :
3.037229847750311665115172806783328447911896669343
42259515007060199595805908881107988253347572678975
2872977727062
X10^ -87

erfc
MIMI = :
1.0
X10^ -96

X = :
1.6
X10^ 1

ERFC(x) = :
2.328485751571530693364872854573442597534396948094
94802151649509557369523737790080619238332625392233
1153810817962
X10^ -113

\ ***************************************************

erfc
X = :
1.7
X10^ 1

ERFC(x) = :
1.021228015094260881145599235077652994401543730587
57117255350655550943197600942423095274220475440447
1071116583101
X10^ -127

erfc
X = :
1.8
X10^ 1

ERFC(x) = :
6.082369231816399307668466715702274949587610076867
57685159278589190532744511115561184378302587387186
5094174183213
X10^ -143

erfc
X = :
1.9
X10^ 1

ERFC(x) = :
4.917722839256475446413297625239608170930672249989
25903631521698356756256672056877874774045924679965
6966106818571
X10^ -159

erfc
X = :
2.0
X10^ 1

ERFC(x) = :
5.395865611607900928934999167905345604088272670923
60528347010378491301876315666993333543074240234371
03862266973
X10^ -176

erfc
X = :
2.1
X10^ 1

ERFC(x) = :
8.032453871022455669021356947138268888967875692513
37885278602378401460562630878875250682660497529814
4290532553308
X10^ -194

erfc
X = :
2.5
X10^ 1

ERFC(x) = :
8.300172571196522752044012769513722768714223191886
69959605444858200128083858466509582535907094157923
0337303216692
X10^ -274

erfc
X = :
3.0
X10^ 1

ERFC(x) = :
2.564656203756111600033397277501447146548889722778
61705412259958618423869477919735075745592460023192
5697543021348
X10^ -393

erfc
X = :
4.0
X10^ 1

ERFC(x) = :
1.896961059966276509268278259713415434936907563929
18618346283475290041180520511188660525669077676004
1365306022708
X10^ -697

erfc
X = :
5.0
X10^ 1

ERFC(x) = :
2.070920778841656048448447875165788792932250920995
39968376623553375989816942513882393211712997100144
289795083031
X10^ -1088

erfc
X = :
1.0
X10^ 2

ERFC(x) = :
6.405961424921732039021339148586394148214414399460
33805776710765024890255482950583122794586657987149
7097903402585
X10^ -4346

erfc
X = :
2.0
X10^ 2

ERFC(x) = :
4.689359295983601498035699776992316770422384630544
86566387750769583162476707139525208321549676277950
7240566295404
X10^ -17375

\ Testing was stopped at x = 200.
\ If x>200, every x must take a long period of time for testing, owing to the exp(x*x) evaluation when x*x >  40000.

(4). Acknowledgement  

These days, I am very sad for the disappearance of FSL (Forth Scientific Library) website. This is the first time I have the opportunity to say many thanks to all contributors of FSL in English. In my Chinese posted articles, I was always to introduce Chinese all the FSL program contributors and to say thanks in Chinese for FSL to all readers. I gave the name “FORTH sages and able man of the past” to all contributors of FSL, they helped to build Forth Scientific Library. And so let me easy to develop my ABC FORTH. Even now this system has not been finished yet. I would like to say thanks first to some key FORTH sages for this article.  

Thanks to:
Leonard Francis Zettel, his big number program “Forth Scientific Library Algorithm #47” had been used in my system.

There are many very important persons else I have to say thanks to them, especially, all contributors of Win32Forth system. And if let us get back to the early years, Both Forth sages Charles H. Moore and Michael Perry, had to be included also. Their code “Tiny BASIC compiler” had been used in my system as well, and it was the fundamentals of my ABC FORTH system.

I believe. You are coming here for the technical reasons not for English. And I prefer to practice FORTH coding rather than to prepare English article. I posted this article by way of Tom Zimmer’s editor WinEd first, reviewed it on site later. It is inevitable to modify something of this article from time to time.

2013/03/17 modified
2016/06/09 modified

\ ******************* THE END ********************