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

Average Error: 19.3 → 0.1

Time: 7.8s

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 r457948 = x;
        double r457949 = y;
        double r457950 = z;
        double r457951 = 0.0692910599291889;
        double r457952 = r457950 * r457951;
        double r457953 = 0.4917317610505968;
        double r457954 = r457952 + r457953;
        double r457955 = r457954 * r457950;
        double r457956 = 0.279195317918525;
        double r457957 = r457955 + r457956;
        double r457958 = r457949 * r457957;
        double r457959 = 6.012459259764103;
        double r457960 = r457950 + r457959;
        double r457961 = r457960 * r457950;
        double r457962 = 3.350343815022304;
        double r457963 = r457961 + r457962;
        double r457964 = r457958 / r457963;
        double r457965 = r457948 + r457964;
        return r457965;
}

↓

double f(double x, double y, double z) {
        double r457966 = z;
        double r457967 = -2712764.504135835;
        bool r457968 = r457966 <= r457967;
        double r457969 = 63485.56364381746;
        bool r457970 = r457966 <= r457969;
        double r457971 = !r457970;
        bool r457972 = r457968 || r457971;
        double r457973 = x;
        double r457974 = y;
        double r457975 = 0.07512208616047561;
        double r457976 = r457975 / r457966;
        double r457977 = 0.0692910599291889;
        double r457978 = 0.40462203869992125;
        double r457979 = 2.0;
        double r457980 = pow(r457966, r457979);
        double r457981 = r457978 / r457980;
        double r457982 = r457977 - r457981;
        double r457983 = r457976 + r457982;
        double r457984 = r457974 * r457983;
        double r457985 = r457973 + r457984;
        double r457986 = r457966 * r457977;
        double r457987 = 0.4917317610505968;
        double r457988 = r457986 + r457987;
        double r457989 = cbrt(r457988);
        double r457990 = r457989 * r457989;
        double r457991 = r457989 * r457966;
        double r457992 = r457990 * r457991;
        double r457993 = 0.279195317918525;
        double r457994 = r457992 + r457993;
        double r457995 = 6.012459259764103;
        double r457996 = r457966 + r457995;
        double r457997 = r457996 * r457966;
        double r457998 = 3.350343815022304;
        double r457999 = r457997 + r457998;
        double r458000 = r457994 / r457999;
        double r458001 = r457974 * r458000;
        double r458002 = r457973 + r458001;
        double r458003 = r457972 ? r457985 : r458002;
        return r458003;
}

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))))