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

Average Error: 26.7 → 0.8

Time: 23.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -394748133220946787888436044907157651456:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)\\ \mathbf{elif}\;x \le 998303336278827940708352:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) - 263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645\right) \cdot x}{\left(x + 43.3400022514000013984514225739985704422\right) \cdot x - 263.5050747210000281484099105000495910645} + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r269517 = x;
        double r269518 = 2.0;
        double r269519 = r269517 - r269518;
        double r269520 = 4.16438922228;
        double r269521 = r269517 * r269520;
        double r269522 = 78.6994924154;
        double r269523 = r269521 + r269522;
        double r269524 = r269523 * r269517;
        double r269525 = 137.519416416;
        double r269526 = r269524 + r269525;
        double r269527 = r269526 * r269517;
        double r269528 = y;
        double r269529 = r269527 + r269528;
        double r269530 = r269529 * r269517;
        double r269531 = z;
        double r269532 = r269530 + r269531;
        double r269533 = r269519 * r269532;
        double r269534 = 43.3400022514;
        double r269535 = r269517 + r269534;
        double r269536 = r269535 * r269517;
        double r269537 = 263.505074721;
        double r269538 = r269536 + r269537;
        double r269539 = r269538 * r269517;
        double r269540 = 313.399215894;
        double r269541 = r269539 + r269540;
        double r269542 = r269541 * r269517;
        double r269543 = 47.066876606;
        double r269544 = r269542 + r269543;
        double r269545 = r269533 / r269544;
        return r269545;
}

↓

double f(double x, double y, double z) {
        double r269546 = x;
        double r269547 = -3.947481332209468e+38;
        bool r269548 = r269546 <= r269547;
        double r269549 = 2.0;
        double r269550 = r269546 - r269549;
        double r269551 = y;
        double r269552 = 3.0;
        double r269553 = pow(r269546, r269552);
        double r269554 = r269551 / r269553;
        double r269555 = 4.16438922228;
        double r269556 = r269554 + r269555;
        double r269557 = 101.7851458539211;
        double r269558 = r269557 / r269546;
        double r269559 = r269556 - r269558;
        double r269560 = r269550 * r269559;
        double r269561 = 9.98303336278828e+23;
        bool r269562 = r269546 <= r269561;
        double r269563 = r269546 * r269555;
        double r269564 = 78.6994924154;
        double r269565 = r269563 + r269564;
        double r269566 = r269565 * r269546;
        double r269567 = 137.519416416;
        double r269568 = r269566 + r269567;
        double r269569 = r269568 * r269546;
        double r269570 = r269569 + r269551;
        double r269571 = r269570 * r269546;
        double r269572 = z;
        double r269573 = r269571 + r269572;
        double r269574 = 43.3400022514;
        double r269575 = r269546 + r269574;
        double r269576 = r269575 * r269546;
        double r269577 = r269576 * r269576;
        double r269578 = 263.505074721;
        double r269579 = r269578 * r269578;
        double r269580 = r269577 - r269579;
        double r269581 = r269580 * r269546;
        double r269582 = r269576 - r269578;
        double r269583 = r269581 / r269582;
        double r269584 = 313.399215894;
        double r269585 = r269583 + r269584;
        double r269586 = r269585 * r269546;
        double r269587 = 47.066876606;
        double r269588 = r269586 + r269587;
        double r269589 = r269573 / r269588;
        double r269590 = r269550 * r269589;
        double r269591 = 2.0;
        double r269592 = pow(r269546, r269591);
        double r269593 = r269551 / r269592;
        double r269594 = r269555 * r269546;
        double r269595 = r269593 + r269594;
        double r269596 = 110.1139242984811;
        double r269597 = r269595 - r269596;
        double r269598 = r269562 ? r269590 : r269597;
        double r269599 = r269548 ? r269560 : r269598;
        return r269599;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.7
Target	0.6
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if x < -3.947481332209468e+38

   1. Initial program 60.4

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

   3. Applied *-un-lft-identity 60.4

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

   4. Applied times-frac 56.5

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

   5. Simplified 56.5

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

   6. Taylor expanded around inf 0.7

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - 101.785145853921093817007204052060842514 \cdot \frac{1}{x}\right)}\]

   7. Simplified 0.7

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)}\]

   if -3.947481332209468e+38 < x < 9.98303336278828e+23

   1. Initial program 0.7

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

   3. Applied *-un-lft-identity 0.7

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

   4. Applied times-frac 0.3

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

   5. Simplified 0.3

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

   7. Applied flip-+ 0.3

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

   8. Applied associate-*l/ 0.3

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

   if 9.98303336278828e+23 < x

   1. Initial program 56.5

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

   2. Taylor expanded around inf 2.2

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -394748133220946787888436044907157651456:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)\\ \mathbf{elif}\;x \le 998303336278827940708352:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) - 263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645\right) \cdot x}{\left(x + 43.3400022514000013984514225739985704422\right) \cdot x - 263.5050747210000281484099105000495910645} + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))