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

Average Error: 26.6 → 1.0

Time: 8.0s

Precision: 64

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.63504100427729212 \cdot 10^{34}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;x \le 1.6132622111277306 \cdot 10^{68}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r430351 = x;
        double r430352 = 2.0;
        double r430353 = r430351 - r430352;
        double r430354 = 4.16438922228;
        double r430355 = r430351 * r430354;
        double r430356 = 78.6994924154;
        double r430357 = r430355 + r430356;
        double r430358 = r430357 * r430351;
        double r430359 = 137.519416416;
        double r430360 = r430358 + r430359;
        double r430361 = r430360 * r430351;
        double r430362 = y;
        double r430363 = r430361 + r430362;
        double r430364 = r430363 * r430351;
        double r430365 = z;
        double r430366 = r430364 + r430365;
        double r430367 = r430353 * r430366;
        double r430368 = 43.3400022514;
        double r430369 = r430351 + r430368;
        double r430370 = r430369 * r430351;
        double r430371 = 263.505074721;
        double r430372 = r430370 + r430371;
        double r430373 = r430372 * r430351;
        double r430374 = 313.399215894;
        double r430375 = r430373 + r430374;
        double r430376 = r430375 * r430351;
        double r430377 = 47.066876606;
        double r430378 = r430376 + r430377;
        double r430379 = r430367 / r430378;
        return r430379;
}

↓

double f(double x, double y, double z) {
        double r430380 = x;
        double r430381 = -2.635041004277292e+34;
        bool r430382 = r430380 <= r430381;
        double r430383 = 2.0;
        double r430384 = r430380 - r430383;
        double r430385 = y;
        double r430386 = 3.0;
        double r430387 = pow(r430380, r430386);
        double r430388 = r430385 / r430387;
        double r430389 = 4.16438922228;
        double r430390 = r430388 + r430389;
        double r430391 = 101.7851458539211;
        double r430392 = 1.0;
        double r430393 = r430392 / r430380;
        double r430394 = r430391 * r430393;
        double r430395 = r430390 - r430394;
        double r430396 = r430384 * r430395;
        double r430397 = 1.6132622111277306e+68;
        bool r430398 = r430380 <= r430397;
        double r430399 = 43.3400022514;
        double r430400 = r430380 + r430399;
        double r430401 = r430400 * r430380;
        double r430402 = 263.505074721;
        double r430403 = r430401 + r430402;
        double r430404 = r430403 * r430380;
        double r430405 = 313.399215894;
        double r430406 = r430404 + r430405;
        double r430407 = r430406 * r430380;
        double r430408 = 47.066876606;
        double r430409 = r430407 + r430408;
        double r430410 = sqrt(r430409);
        double r430411 = r430392 / r430410;
        double r430412 = r430380 * r430389;
        double r430413 = 78.6994924154;
        double r430414 = r430412 + r430413;
        double r430415 = r430414 * r430380;
        double r430416 = 137.519416416;
        double r430417 = r430415 + r430416;
        double r430418 = r430417 * r430380;
        double r430419 = r430418 + r430385;
        double r430420 = r430419 * r430380;
        double r430421 = z;
        double r430422 = r430420 + r430421;
        double r430423 = r430422 / r430410;
        double r430424 = r430411 * r430423;
        double r430425 = r430384 * r430424;
        double r430426 = 2.0;
        double r430427 = pow(r430380, r430426);
        double r430428 = r430385 / r430427;
        double r430429 = r430389 * r430380;
        double r430430 = r430428 + r430429;
        double r430431 = 110.1139242984811;
        double r430432 = r430430 - r430431;
        double r430433 = r430398 ? r430425 : r430432;
        double r430434 = r430382 ? r430396 : r430433;
        return r430434;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.6
Target	0.5
Herbie	1.0

\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -2.635041004277292e+34

   1. Initial program 59.1

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 59.1

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001\right)}}\]

   4. Applied times-frac 54.8

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\]

   5. Simplified 54.8

      \[\leadsto \color{blue}{\left(x - 2\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

   6. Taylor expanded around inf 1.6

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)}\]

   if -2.635041004277292e+34 < x < 1.6132622111277306e+68

   1. Initial program 2.0

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 2.0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001\right)}}\]

   4. Applied times-frac 0.6

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\]

   5. Simplified 0.6

      \[\leadsto \color{blue}{\left(x - 2\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 0.8

      \[\leadsto \left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\color{blue}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001} \cdot \sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}}\]

   8. Applied *-un-lft-identity 0.8

      \[\leadsto \left(x - 2\right) \cdot \frac{\color{blue}{1 \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001} \cdot \sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\]

   9. Applied times-frac 1.0

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\right)}\]

   if 1.6132622111277306e+68 < x

   1. Initial program 64.0

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

   2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.63504100427729212 \cdot 10^{34}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;x \le 1.6132622111277306 \cdot 10^{68}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* 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) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))