Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C

Average Error: 27.4 → 0.9

Time: 33.0s

Precision: 64

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.54130501806106185525358024175399241783 \cdot 10^{76}:\\ \;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{\frac{y}{x}}{x} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 291648759279740435667263883684208451780600:\\ \;\;\;\;\frac{\left(\left(\left(\left(137.5194164160000127594685181975364685059 + x \cdot \left(78.69949241540000173245061887428164482117 + 4.16438922227999963610045597306452691555 \cdot x\right)\right) \cdot x + y\right) \cdot x + z\right) \cdot \frac{x - 2}{\sqrt{\sqrt{47.06687660600000100430406746454536914825 + \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(43.3400022514000013984514225739985704422 + x\right) \cdot x\right)\right) \cdot x}}}\right) \cdot \frac{1}{\sqrt{\sqrt{47.06687660600000100430406746454536914825 + \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(43.3400022514000013984514225739985704422 + x\right) \cdot x\right)\right) \cdot x}}}}{\sqrt{47.06687660600000100430406746454536914825 + \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(43.3400022514000013984514225739985704422 + x\right) \cdot x\right)\right) \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{\frac{y}{x}}{x} - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r20585361 = x;
        double r20585362 = 2.0;
        double r20585363 = r20585361 - r20585362;
        double r20585364 = 4.16438922228;
        double r20585365 = r20585361 * r20585364;
        double r20585366 = 78.6994924154;
        double r20585367 = r20585365 + r20585366;
        double r20585368 = r20585367 * r20585361;
        double r20585369 = 137.519416416;
        double r20585370 = r20585368 + r20585369;
        double r20585371 = r20585370 * r20585361;
        double r20585372 = y;
        double r20585373 = r20585371 + r20585372;
        double r20585374 = r20585373 * r20585361;
        double r20585375 = z;
        double r20585376 = r20585374 + r20585375;
        double r20585377 = r20585363 * r20585376;
        double r20585378 = 43.3400022514;
        double r20585379 = r20585361 + r20585378;
        double r20585380 = r20585379 * r20585361;
        double r20585381 = 263.505074721;
        double r20585382 = r20585380 + r20585381;
        double r20585383 = r20585382 * r20585361;
        double r20585384 = 313.399215894;
        double r20585385 = r20585383 + r20585384;
        double r20585386 = r20585385 * r20585361;
        double r20585387 = 47.066876606;
        double r20585388 = r20585386 + r20585387;
        double r20585389 = r20585377 / r20585388;
        return r20585389;
}

↓

double f(double x, double y, double z) {
        double r20585390 = x;
        double r20585391 = -4.541305018061062e+76;
        bool r20585392 = r20585390 <= r20585391;
        double r20585393 = 4.16438922228;
        double r20585394 = r20585393 * r20585390;
        double r20585395 = y;
        double r20585396 = r20585395 / r20585390;
        double r20585397 = r20585396 / r20585390;
        double r20585398 = 110.1139242984811;
        double r20585399 = r20585397 - r20585398;
        double r20585400 = r20585394 + r20585399;
        double r20585401 = 2.9164875927974044e+41;
        bool r20585402 = r20585390 <= r20585401;
        double r20585403 = 137.519416416;
        double r20585404 = 78.6994924154;
        double r20585405 = r20585404 + r20585394;
        double r20585406 = r20585390 * r20585405;
        double r20585407 = r20585403 + r20585406;
        double r20585408 = r20585407 * r20585390;
        double r20585409 = r20585408 + r20585395;
        double r20585410 = r20585409 * r20585390;
        double r20585411 = z;
        double r20585412 = r20585410 + r20585411;
        double r20585413 = 2.0;
        double r20585414 = r20585390 - r20585413;
        double r20585415 = 47.066876606;
        double r20585416 = 313.399215894;
        double r20585417 = 263.505074721;
        double r20585418 = 43.3400022514;
        double r20585419 = r20585418 + r20585390;
        double r20585420 = r20585419 * r20585390;
        double r20585421 = r20585417 + r20585420;
        double r20585422 = r20585390 * r20585421;
        double r20585423 = r20585416 + r20585422;
        double r20585424 = r20585423 * r20585390;
        double r20585425 = r20585415 + r20585424;
        double r20585426 = sqrt(r20585425);
        double r20585427 = sqrt(r20585426);
        double r20585428 = r20585414 / r20585427;
        double r20585429 = r20585412 * r20585428;
        double r20585430 = 1.0;
        double r20585431 = r20585430 / r20585427;
        double r20585432 = r20585429 * r20585431;
        double r20585433 = r20585432 / r20585426;
        double r20585434 = r20585402 ? r20585433 : r20585400;
        double r20585435 = r20585392 ? r20585400 : r20585434;
        return r20585435;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.4
Target	0.6
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -4.541305018061062e+76 or 2.9164875927974044e+41 < x

   1. Initial program 61.8

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

   2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229}\]

   3. Simplified 0.5

      \[\leadsto \color{blue}{4.16438922227999963610045597306452691555 \cdot x + \left(\frac{\frac{y}{x}}{x} - 110.1139242984810948655649553984403610229\right)}\]

   if -4.541305018061062e+76 < x < 2.9164875927974044e+41

   1. Initial program 2.8

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 3.0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825} \cdot \sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}}\]

   4. Applied times-frac 1.2

      \[\leadsto \color{blue}{\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 1.2

      \[\leadsto \frac{x - 2}{\sqrt{\color{blue}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825} \cdot \sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\]

   7. Applied sqrt-prod 1.6

      \[\leadsto \frac{x - 2}{\color{blue}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}} \cdot \sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\]

   8. Applied *-un-lft-identity 1.6

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x - 2\right)}}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}} \cdot \sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\]

   9. Applied times-frac 1.2

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}} \cdot \frac{x - 2}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}}\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\]

   10. Applied associate-*l* 1.2

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}} \cdot \left(\frac{x - 2}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\right)}\]
   11. Using strategy rm

   12. Applied associate-*r/ 1.2

       \[\leadsto \frac{1}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}} \cdot \color{blue}{\frac{\frac{x - 2}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}} \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}}\]

   13. Applied associate-*r/ 1.2

       \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}} \cdot \left(\frac{x - 2}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}} \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)\right)}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.54130501806106185525358024175399241783 \cdot 10^{76}:\\ \;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{\frac{y}{x}}{x} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 291648759279740435667263883684208451780600:\\ \;\;\;\;\frac{\left(\left(\left(\left(137.5194164160000127594685181975364685059 + x \cdot \left(78.69949241540000173245061887428164482117 + 4.16438922227999963610045597306452691555 \cdot x\right)\right) \cdot x + y\right) \cdot x + z\right) \cdot \frac{x - 2}{\sqrt{\sqrt{47.06687660600000100430406746454536914825 + \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(43.3400022514000013984514225739985704422 + x\right) \cdot x\right)\right) \cdot x}}}\right) \cdot \frac{1}{\sqrt{\sqrt{47.06687660600000100430406746454536914825 + \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(43.3400022514000013984514225739985704422 + x\right) \cdot x\right)\right) \cdot x}}}}{\sqrt{47.06687660600000100430406746454536914825 + \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(43.3400022514000013984514225739985704422 + x\right) \cdot x\right)\right) \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{\frac{y}{x}}{x} - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1.0) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))