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

Average Error: 19.3 → 0.1

Time: 7.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2712764.504135835 \lor \neg \left(z \le 63485.5636438174624\right):\\ \;\;\;\;x + y \cdot \left(\frac{0.07512208616047561}{z} + \left(0.0692910599291888946 - \frac{0.404622038699921249}{{z}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right) + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r174753 = x;
        double r174754 = y;
        double r174755 = z;
        double r174756 = 0.0692910599291889;
        double r174757 = r174755 * r174756;
        double r174758 = 0.4917317610505968;
        double r174759 = r174757 + r174758;
        double r174760 = r174759 * r174755;
        double r174761 = 0.279195317918525;
        double r174762 = r174760 + r174761;
        double r174763 = r174754 * r174762;
        double r174764 = 6.012459259764103;
        double r174765 = r174755 + r174764;
        double r174766 = r174765 * r174755;
        double r174767 = 3.350343815022304;
        double r174768 = r174766 + r174767;
        double r174769 = r174763 / r174768;
        double r174770 = r174753 + r174769;
        return r174770;
}

↓

double f(double x, double y, double z) {
        double r174771 = z;
        double r174772 = -2712764.504135835;
        bool r174773 = r174771 <= r174772;
        double r174774 = 63485.56364381746;
        bool r174775 = r174771 <= r174774;
        double r174776 = !r174775;
        bool r174777 = r174773 || r174776;
        double r174778 = x;
        double r174779 = y;
        double r174780 = 0.07512208616047561;
        double r174781 = r174780 / r174771;
        double r174782 = 0.0692910599291889;
        double r174783 = 0.40462203869992125;
        double r174784 = 2.0;
        double r174785 = pow(r174771, r174784);
        double r174786 = r174783 / r174785;
        double r174787 = r174782 - r174786;
        double r174788 = r174781 + r174787;
        double r174789 = r174779 * r174788;
        double r174790 = r174778 + r174789;
        double r174791 = r174771 * r174782;
        double r174792 = 0.4917317610505968;
        double r174793 = r174791 + r174792;
        double r174794 = cbrt(r174793);
        double r174795 = r174794 * r174794;
        double r174796 = r174794 * r174771;
        double r174797 = r174795 * r174796;
        double r174798 = 0.279195317918525;
        double r174799 = r174797 + r174798;
        double r174800 = 6.012459259764103;
        double r174801 = r174771 + r174800;
        double r174802 = r174801 * r174771;
        double r174803 = 3.350343815022304;
        double r174804 = r174802 + r174803;
        double r174805 = r174799 / r174804;
        double r174806 = r174779 * r174805;
        double r174807 = r174778 + r174806;
        double r174808 = r174777 ? r174790 : r174807;
        return r174808;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.3
Target	0.2
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -2712764.504135835 or 63485.56364381746 < z

   1. Initial program 39.5

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 39.5

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\color{blue}{1 \cdot \left(\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394\right)}}\]

   4. Applied times-frac 31.5

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\]

   5. Simplified 31.5

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 31.7

      \[\leadsto x + y \cdot \frac{\color{blue}{\left(\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right)} \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

   8. Applied associate-*l* 31.7

      \[\leadsto x + y \cdot \frac{\color{blue}{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right)} + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

   9. Taylor expanded around inf 0.0

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.07512208616047561 \cdot \frac{1}{z} + 0.0692910599291888946\right) - 0.404622038699921249 \cdot \frac{1}{{z}^{2}}\right)}\]

   10. Simplified 0.0

       \[\leadsto x + y \cdot \color{blue}{\left(\frac{0.07512208616047561}{z} + \left(0.0692910599291888946 - \frac{0.404622038699921249}{{z}^{2}}\right)\right)}\]

   if -2712764.504135835 < z < 63485.56364381746

   1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.2

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\color{blue}{1 \cdot \left(\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394\right)}}\]

   4. Applied times-frac 0.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\]

   5. Simplified 0.1

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 0.1

      \[\leadsto x + y \cdot \frac{\color{blue}{\left(\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right)} \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

   8. Applied associate-*l* 0.1

      \[\leadsto x + y \cdot \frac{\color{blue}{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right)} + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2712764.504135835 \lor \neg \left(z \le 63485.5636438174624\right):\\ \;\;\;\;x + y \cdot \left(\frac{0.07512208616047561}{z} + \left(0.0692910599291888946 - \frac{0.404622038699921249}{{z}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right) + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))