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

Average Error: 27.1 → 0.8

Time: 7.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.734460045413716797390213553569907175702 \cdot 10^{73} \lor \neg \left(x \le 1.945381819008962171531860479288483682136 \cdot 10^{55}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r394551 = x;
        double r394552 = 2.0;
        double r394553 = r394551 - r394552;
        double r394554 = 4.16438922228;
        double r394555 = r394551 * r394554;
        double r394556 = 78.6994924154;
        double r394557 = r394555 + r394556;
        double r394558 = r394557 * r394551;
        double r394559 = 137.519416416;
        double r394560 = r394558 + r394559;
        double r394561 = r394560 * r394551;
        double r394562 = y;
        double r394563 = r394561 + r394562;
        double r394564 = r394563 * r394551;
        double r394565 = z;
        double r394566 = r394564 + r394565;
        double r394567 = r394553 * r394566;
        double r394568 = 43.3400022514;
        double r394569 = r394551 + r394568;
        double r394570 = r394569 * r394551;
        double r394571 = 263.505074721;
        double r394572 = r394570 + r394571;
        double r394573 = r394572 * r394551;
        double r394574 = 313.399215894;
        double r394575 = r394573 + r394574;
        double r394576 = r394575 * r394551;
        double r394577 = 47.066876606;
        double r394578 = r394576 + r394577;
        double r394579 = r394567 / r394578;
        return r394579;
}

↓

double f(double x, double y, double z) {
        double r394580 = x;
        double r394581 = -8.734460045413717e+73;
        bool r394582 = r394580 <= r394581;
        double r394583 = 1.9453818190089622e+55;
        bool r394584 = r394580 <= r394583;
        double r394585 = !r394584;
        bool r394586 = r394582 || r394585;
        double r394587 = y;
        double r394588 = 2.0;
        double r394589 = pow(r394580, r394588);
        double r394590 = r394587 / r394589;
        double r394591 = 4.16438922228;
        double r394592 = r394591 * r394580;
        double r394593 = r394590 + r394592;
        double r394594 = 110.1139242984811;
        double r394595 = r394593 - r394594;
        double r394596 = 2.0;
        double r394597 = r394580 - r394596;
        double r394598 = 1.0;
        double r394599 = 43.3400022514;
        double r394600 = r394580 + r394599;
        double r394601 = r394600 * r394580;
        double r394602 = 263.505074721;
        double r394603 = r394601 + r394602;
        double r394604 = r394603 * r394580;
        double r394605 = 313.399215894;
        double r394606 = r394604 + r394605;
        double r394607 = r394606 * r394580;
        double r394608 = 47.066876606;
        double r394609 = r394607 + r394608;
        double r394610 = sqrt(r394609);
        double r394611 = r394598 / r394610;
        double r394612 = r394580 * r394591;
        double r394613 = 78.6994924154;
        double r394614 = r394612 + r394613;
        double r394615 = r394614 * r394580;
        double r394616 = 137.519416416;
        double r394617 = r394615 + r394616;
        double r394618 = r394617 * r394580;
        double r394619 = r394618 + r394587;
        double r394620 = r394619 * r394580;
        double r394621 = z;
        double r394622 = r394620 + r394621;
        double r394623 = r394622 / r394610;
        double r394624 = r394611 * r394623;
        double r394625 = r394597 * r394624;
        double r394626 = r394586 ? r394595 : r394625;
        return r394626;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.1
Target	0.5
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 2 regimes

2. if x < -8.734460045413717e+73 or 1.9453818190089622e+55 < x

   1. Initial program 63.5

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

   2. Taylor expanded around inf 0.2

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

   if -8.734460045413717e+73 < x < 1.9453818190089622e+55

   1. Initial program 2.5

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

   3. Applied *-un-lft-identity 2.5

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

   4. Applied times-frac 0.8

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

   5. Simplified 0.8

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

   7. Applied add-sqr-sqrt 1.0

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

   8. Applied *-un-lft-identity 1.0

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

   9. Applied times-frac 1.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.734460045413716797390213553569907175702 \cdot 10^{73} \lor \neg \left(x \le 1.945381819008962171531860479288483682136 \cdot 10^{55}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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)))