Average Error: 20.2 → 0.1
Time: 1.1m
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.65591962832949798 \cdot 10^{31} \lor \neg \left(z \le 856589.856180236093\right):\\ \;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977} \cdot \sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}{\frac{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}}\\ \end{array}\]
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 -6.65591962832949798 \cdot 10^{31} \lor \neg \left(z \le 856589.856180236093\right):\\
\;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977} \cdot \sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}{\frac{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}}\\

\end{array}
double f(double x, double y, double z) {
        double r332046 = x;
        double r332047 = y;
        double r332048 = z;
        double r332049 = 0.0692910599291889;
        double r332050 = r332048 * r332049;
        double r332051 = 0.4917317610505968;
        double r332052 = r332050 + r332051;
        double r332053 = r332052 * r332048;
        double r332054 = 0.279195317918525;
        double r332055 = r332053 + r332054;
        double r332056 = r332047 * r332055;
        double r332057 = 6.012459259764103;
        double r332058 = r332048 + r332057;
        double r332059 = r332058 * r332048;
        double r332060 = 3.350343815022304;
        double r332061 = r332059 + r332060;
        double r332062 = r332056 / r332061;
        double r332063 = r332046 + r332062;
        return r332063;
}
↓
double f(double x, double y, double z) {
        double r332064 = z;
        double r332065 = -6.655919628329498e+31;
        bool r332066 = r332064 <= r332065;
        double r332067 = 856589.8561802361;
        bool r332068 = r332064 <= r332067;
        double r332069 = !r332068;
        bool r332070 = r332066 || r332069;
        double r332071 = x;
        double r332072 = 0.07512208616047561;
        double r332073 = y;
        double r332074 = r332073 / r332064;
        double r332075 = r332072 * r332074;
        double r332076 = 0.0692910599291889;
        double r332077 = r332076 * r332073;
        double r332078 = r332075 + r332077;
        double r332079 = r332071 + r332078;
        double r332080 = r332064 * r332076;
        double r332081 = 0.4917317610505968;
        double r332082 = r332080 + r332081;
        double r332083 = r332082 * r332064;
        double r332084 = 0.279195317918525;
        double r332085 = r332083 + r332084;
        double r332086 = cbrt(r332085);
        double r332087 = r332086 * r332086;
        double r332088 = 6.012459259764103;
        double r332089 = r332064 + r332088;
        double r332090 = r332089 * r332064;
        double r332091 = 3.350343815022304;
        double r332092 = r332090 + r332091;
        double r332093 = r332092 / r332086;
        double r332094 = r332087 / r332093;
        double r332095 = r332073 * r332094;
        double r332096 = r332071 + r332095;
        double r332097 = r332070 ? r332079 : r332096;
        return r332097;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.655919628329498e+31 or 856589.8561802361 < z

    1. Initial program 42.8

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

    2. Taylor expanded around inf 0.0

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

    if -6.655919628329498e+31 < z < 856589.8561802361

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

    7. Applied add-cube-cbrt0.2

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

    8. Applied associate-/l*0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.65591962832949798 \cdot 10^{31} \lor \neg \left(z \le 856589.856180236093\right):\\ \;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977} \cdot \sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}{\frac{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1.0 (+ (* (+ 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))))