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

Average Error: 25.4 → 0.8

Time: 28.9s

Precision: 64

Math

\[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.508874372053853 \cdot 10^{+28}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \mathbf{elif}\;x \le 9.203477273753945 \cdot 10^{+42}:\\ \;\;\;\;\frac{z + x \cdot \left(y + \left(x \cdot \left(78.6994924154 + 4.16438922228 \cdot x\right) + 137.519416416\right) \cdot x\right)}{\sqrt{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}} \cdot \left(\frac{1}{\sqrt{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}} \cdot \frac{x - 2.0}{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r22146527 = x;
        double r22146528 = 2.0;
        double r22146529 = r22146527 - r22146528;
        double r22146530 = 4.16438922228;
        double r22146531 = r22146527 * r22146530;
        double r22146532 = 78.6994924154;
        double r22146533 = r22146531 + r22146532;
        double r22146534 = r22146533 * r22146527;
        double r22146535 = 137.519416416;
        double r22146536 = r22146534 + r22146535;
        double r22146537 = r22146536 * r22146527;
        double r22146538 = y;
        double r22146539 = r22146537 + r22146538;
        double r22146540 = r22146539 * r22146527;
        double r22146541 = z;
        double r22146542 = r22146540 + r22146541;
        double r22146543 = r22146529 * r22146542;
        double r22146544 = 43.3400022514;
        double r22146545 = r22146527 + r22146544;
        double r22146546 = r22146545 * r22146527;
        double r22146547 = 263.505074721;
        double r22146548 = r22146546 + r22146547;
        double r22146549 = r22146548 * r22146527;
        double r22146550 = 313.399215894;
        double r22146551 = r22146549 + r22146550;
        double r22146552 = r22146551 * r22146527;
        double r22146553 = 47.066876606;
        double r22146554 = r22146552 + r22146553;
        double r22146555 = r22146543 / r22146554;
        return r22146555;
}

↓

double f(double x, double y, double z) {
        double r22146556 = x;
        double r22146557 = -2.508874372053853e+28;
        bool r22146558 = r22146556 <= r22146557;
        double r22146559 = 4.16438922228;
        double r22146560 = r22146559 * r22146556;
        double r22146561 = y;
        double r22146562 = r22146556 * r22146556;
        double r22146563 = r22146561 / r22146562;
        double r22146564 = 110.1139242984811;
        double r22146565 = r22146563 - r22146564;
        double r22146566 = r22146560 + r22146565;
        double r22146567 = 9.203477273753945e+42;
        bool r22146568 = r22146556 <= r22146567;
        double r22146569 = z;
        double r22146570 = 78.6994924154;
        double r22146571 = r22146570 + r22146560;
        double r22146572 = r22146556 * r22146571;
        double r22146573 = 137.519416416;
        double r22146574 = r22146572 + r22146573;
        double r22146575 = r22146574 * r22146556;
        double r22146576 = r22146561 + r22146575;
        double r22146577 = r22146556 * r22146576;
        double r22146578 = r22146569 + r22146577;
        double r22146579 = 47.066876606;
        double r22146580 = 43.3400022514;
        double r22146581 = r22146580 + r22146556;
        double r22146582 = r22146556 * r22146581;
        double r22146583 = 263.505074721;
        double r22146584 = r22146582 + r22146583;
        double r22146585 = r22146556 * r22146584;
        double r22146586 = 313.399215894;
        double r22146587 = r22146585 + r22146586;
        double r22146588 = r22146556 * r22146587;
        double r22146589 = r22146579 + r22146588;
        double r22146590 = sqrt(r22146589);
        double r22146591 = sqrt(r22146590);
        double r22146592 = r22146578 / r22146591;
        double r22146593 = 1.0;
        double r22146594 = r22146593 / r22146591;
        double r22146595 = 2.0;
        double r22146596 = r22146556 - r22146595;
        double r22146597 = r22146596 / r22146590;
        double r22146598 = r22146594 * r22146597;
        double r22146599 = r22146592 * r22146598;
        double r22146600 = r22146568 ? r22146599 : r22146566;
        double r22146601 = r22146558 ? r22146566 : r22146600;
        return r22146601;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	25.4
Target	0.4
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x \lt 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -2.508874372053853e+28 or 9.203477273753945e+42 < x

   1. Initial program 57.5

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

   2. Taylor expanded around inf 1.1

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922228 \cdot x\right) - 110.1139242984811}\]

   3. Simplified 1.1

      \[\leadsto \color{blue}{4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)}\]

   if -2.508874372053853e+28 < x < 9.203477273753945e+42

   1. Initial program 0.6

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 0.7

      \[\leadsto \frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \cdot \sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}\]

   4. Applied times-frac 0.7

      \[\leadsto \color{blue}{\frac{x - 2.0}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 0.7

      \[\leadsto \frac{x - 2.0}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\sqrt{\color{blue}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \cdot \sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}}\]

   7. Applied sqrt-prod 0.8

      \[\leadsto \frac{x - 2.0}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\color{blue}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \cdot \sqrt{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}}\]

   8. Applied *-un-lft-identity 0.8

      \[\leadsto \frac{x - 2.0}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \cdot \frac{\color{blue}{1 \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \cdot \sqrt{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}\]

   9. Applied times-frac 0.7

      \[\leadsto \frac{x - 2.0}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}\right)}\]

   10. Applied associate-*r* 0.6

       \[\leadsto \color{blue}{\left(\frac{x - 2.0}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \cdot \frac{1}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.508874372053853 \cdot 10^{+28}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \mathbf{elif}\;x \le 9.203477273753945 \cdot 10^{+42}:\\ \;\;\;\;\frac{z + x \cdot \left(y + \left(x \cdot \left(78.6994924154 + 4.16438922228 \cdot x\right) + 137.519416416\right) \cdot x\right)}{\sqrt{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}} \cdot \left(\frac{1}{\sqrt{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}} \cdot \frac{x - 2.0}{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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) (/ (+ (* (+ (* (+ (* (+ (* 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)))