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

Average Error: 26.8 → 0.6

Time: 15.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.053875551376048812700508398730049713438 \cdot 10^{46} \lor \neg \left(x \le 6.241778619282483948639908656517360313067 \cdot 10^{44}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r300534 = x;
        double r300535 = 2.0;
        double r300536 = r300534 - r300535;
        double r300537 = 4.16438922228;
        double r300538 = r300534 * r300537;
        double r300539 = 78.6994924154;
        double r300540 = r300538 + r300539;
        double r300541 = r300540 * r300534;
        double r300542 = 137.519416416;
        double r300543 = r300541 + r300542;
        double r300544 = r300543 * r300534;
        double r300545 = y;
        double r300546 = r300544 + r300545;
        double r300547 = r300546 * r300534;
        double r300548 = z;
        double r300549 = r300547 + r300548;
        double r300550 = r300536 * r300549;
        double r300551 = 43.3400022514;
        double r300552 = r300534 + r300551;
        double r300553 = r300552 * r300534;
        double r300554 = 263.505074721;
        double r300555 = r300553 + r300554;
        double r300556 = r300555 * r300534;
        double r300557 = 313.399215894;
        double r300558 = r300556 + r300557;
        double r300559 = r300558 * r300534;
        double r300560 = 47.066876606;
        double r300561 = r300559 + r300560;
        double r300562 = r300550 / r300561;
        return r300562;
}

↓

double f(double x, double y, double z) {
        double r300563 = x;
        double r300564 = -1.0538755513760488e+46;
        bool r300565 = r300563 <= r300564;
        double r300566 = 6.241778619282484e+44;
        bool r300567 = r300563 <= r300566;
        double r300568 = !r300567;
        bool r300569 = r300565 || r300568;
        double r300570 = y;
        double r300571 = 2.0;
        double r300572 = pow(r300563, r300571);
        double r300573 = r300570 / r300572;
        double r300574 = 4.16438922228;
        double r300575 = r300574 * r300563;
        double r300576 = r300573 + r300575;
        double r300577 = 110.1139242984811;
        double r300578 = r300576 - r300577;
        double r300579 = 2.0;
        double r300580 = r300563 - r300579;
        double r300581 = 43.3400022514;
        double r300582 = r300563 + r300581;
        double r300583 = r300582 * r300563;
        double r300584 = 263.505074721;
        double r300585 = r300583 + r300584;
        double r300586 = r300585 * r300563;
        double r300587 = 313.399215894;
        double r300588 = r300586 + r300587;
        double r300589 = r300588 * r300563;
        double r300590 = 47.066876606;
        double r300591 = r300589 + r300590;
        double r300592 = r300563 * r300574;
        double r300593 = 78.6994924154;
        double r300594 = r300592 + r300593;
        double r300595 = r300594 * r300563;
        double r300596 = 137.519416416;
        double r300597 = r300595 + r300596;
        double r300598 = r300597 * r300563;
        double r300599 = r300598 + r300570;
        double r300600 = r300599 * r300563;
        double r300601 = z;
        double r300602 = r300600 + r300601;
        double r300603 = r300591 / r300602;
        double r300604 = r300580 / r300603;
        double r300605 = r300569 ? r300578 : r300604;
        return r300605;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.8
Target	0.5
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if x < -1.0538755513760488e+46 or 6.241778619282484e+44 < x

   1. Initial program 61.1

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

   2. Taylor expanded around inf 0.5

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

   if -1.0538755513760488e+46 < x < 6.241778619282484e+44

   1. Initial program 1.2

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

   3. Applied associate-/l* 0.6

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.053875551376048812700508398730049713438 \cdot 10^{46} \lor \neg \left(x \le 6.241778619282483948639908656517360313067 \cdot 10^{44}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.3261287258700048e62) (- (+ (/ y (* x x)) (* 4.16438922227999964 x)) 110.11392429848109) (if (< x 9.4299917145546727e55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922227999964) 78.6994924154000017) x) 137.51941641600001) x) y) x) z) (+ (* (+ (+ (* 263.50507472100003 x) (+ (* 43.3400022514000014 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606000001))) (- (+ (/ y (* x x)) (* 4.16438922227999964 x)) 110.11392429848109)))

  (/ (* (- x 2) (+ (* (+ (* (+ (* (+ (* x 4.16438922227999964) 78.6994924154000017) x) 137.51941641600001) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514000014) x) 263.50507472100003) x) 313.399215894) x) 47.066876606000001)))