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

Average Error: 26.4 → 0.9

Time: 26.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -914289206283460288653826229010432 \lor \neg \left(x \le 542737417535511941087232\right):\\ \;\;\;\;\left(x - 2\right) \cdot \left(\frac{y}{{x}^{3}} + \left(4.16438922227999963610045597306452691555 - \frac{101.785145853921093817007204052060842514}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(47.06687660600000100430406746454536914825 - \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)\right) \cdot x\right) \cdot \frac{\frac{z + x \cdot \left(y + \left(137.5194164160000127594685181975364685059 + \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right) \cdot x\right) \cdot x\right)}{\left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)\right) \cdot x + 47.06687660600000100430406746454536914825}}{47.06687660600000100430406746454536914825 - \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)\right) \cdot x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r171639 = x;
        double r171640 = 2.0;
        double r171641 = r171639 - r171640;
        double r171642 = 4.16438922228;
        double r171643 = r171639 * r171642;
        double r171644 = 78.6994924154;
        double r171645 = r171643 + r171644;
        double r171646 = r171645 * r171639;
        double r171647 = 137.519416416;
        double r171648 = r171646 + r171647;
        double r171649 = r171648 * r171639;
        double r171650 = y;
        double r171651 = r171649 + r171650;
        double r171652 = r171651 * r171639;
        double r171653 = z;
        double r171654 = r171652 + r171653;
        double r171655 = r171641 * r171654;
        double r171656 = 43.3400022514;
        double r171657 = r171639 + r171656;
        double r171658 = r171657 * r171639;
        double r171659 = 263.505074721;
        double r171660 = r171658 + r171659;
        double r171661 = r171660 * r171639;
        double r171662 = 313.399215894;
        double r171663 = r171661 + r171662;
        double r171664 = r171663 * r171639;
        double r171665 = 47.066876606;
        double r171666 = r171664 + r171665;
        double r171667 = r171655 / r171666;
        return r171667;
}

↓

double f(double x, double y, double z) {
        double r171668 = x;
        double r171669 = -9.142892062834603e+32;
        bool r171670 = r171668 <= r171669;
        double r171671 = 5.4273741753551194e+23;
        bool r171672 = r171668 <= r171671;
        double r171673 = !r171672;
        bool r171674 = r171670 || r171673;
        double r171675 = 2.0;
        double r171676 = r171668 - r171675;
        double r171677 = y;
        double r171678 = 3.0;
        double r171679 = pow(r171668, r171678);
        double r171680 = r171677 / r171679;
        double r171681 = 4.16438922228;
        double r171682 = 101.7851458539211;
        double r171683 = r171682 / r171668;
        double r171684 = r171681 - r171683;
        double r171685 = r171680 + r171684;
        double r171686 = r171676 * r171685;
        double r171687 = 47.066876606;
        double r171688 = 313.399215894;
        double r171689 = 263.505074721;
        double r171690 = 43.3400022514;
        double r171691 = r171690 + r171668;
        double r171692 = r171668 * r171691;
        double r171693 = r171689 + r171692;
        double r171694 = r171668 * r171693;
        double r171695 = r171688 + r171694;
        double r171696 = r171695 * r171668;
        double r171697 = r171687 - r171696;
        double r171698 = z;
        double r171699 = 137.519416416;
        double r171700 = r171681 * r171668;
        double r171701 = 78.6994924154;
        double r171702 = r171700 + r171701;
        double r171703 = r171702 * r171668;
        double r171704 = r171699 + r171703;
        double r171705 = r171704 * r171668;
        double r171706 = r171677 + r171705;
        double r171707 = r171668 * r171706;
        double r171708 = r171698 + r171707;
        double r171709 = r171696 + r171687;
        double r171710 = r171708 / r171709;
        double r171711 = r171710 / r171697;
        double r171712 = r171697 * r171711;
        double r171713 = r171676 * r171712;
        double r171714 = r171674 ? r171686 : r171713;
        return r171714;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.4
Target	0.5
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if x < -9.142892062834603e+32 or 5.4273741753551194e+23 < x

   1. Initial program 58.1

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

   2. Simplified 53.8

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

   4. Applied div-inv 53.8

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

   5. Applied associate-*l* 53.8

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

   6. Simplified 53.7

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

   7. Taylor expanded around inf 1.6

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

   8. Simplified 1.6

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

   if -9.142892062834603e+32 < x < 5.4273741753551194e+23

   1. Initial program 0.5

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

   2. Simplified 0.6

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

   4. Applied div-inv 0.6

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

   5. Applied associate-*l* 0.6

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

   6. Simplified 0.2

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

   8. Applied flip-+ 1.0

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

   9. Applied associate-/r/ 0.7

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

   10. Simplified 0.3

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -914289206283460288653826229010432 \lor \neg \left(x \le 542737417535511941087232\right):\\ \;\;\;\;\left(x - 2\right) \cdot \left(\frac{y}{{x}^{3}} + \left(4.16438922227999963610045597306452691555 - \frac{101.785145853921093817007204052060842514}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(47.06687660600000100430406746454536914825 - \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)\right) \cdot x\right) \cdot \frac{\frac{z + x \cdot \left(y + \left(137.5194164160000127594685181975364685059 + \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right) \cdot x\right) \cdot x\right)}{\left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)\right) \cdot x + 47.06687660600000100430406746454536914825}}{47.06687660600000100430406746454536914825 - \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)\right) \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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) 1.0) (/ (+ (* (+ (* (+ (* (+ (* 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)))