Average Error: 19.3 → 0.1
Time: 7.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2549507.4500260064 \lor \neg \left(z \le 63485.5636438174624\right):\\ \;\;\;\;x + \left(0.0692910599291888946 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.404622038699921249}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right) + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 -2549507.4500260064 \lor \neg \left(z \le 63485.5636438174624\right):\\
\;\;\;\;x + \left(0.0692910599291888946 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.404622038699921249}{z}\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}
double f(double x, double y, double z) {
        double r396045 = x;
        double r396046 = y;
        double r396047 = z;
        double r396048 = 0.0692910599291889;
        double r396049 = r396047 * r396048;
        double r396050 = 0.4917317610505968;
        double r396051 = r396049 + r396050;
        double r396052 = r396051 * r396047;
        double r396053 = 0.279195317918525;
        double r396054 = r396052 + r396053;
        double r396055 = r396046 * r396054;
        double r396056 = 6.012459259764103;
        double r396057 = r396047 + r396056;
        double r396058 = r396057 * r396047;
        double r396059 = 3.350343815022304;
        double r396060 = r396058 + r396059;
        double r396061 = r396055 / r396060;
        double r396062 = r396045 + r396061;
        return r396062;
}
↓
double f(double x, double y, double z) {
        double r396063 = z;
        double r396064 = -2549507.4500260064;
        bool r396065 = r396063 <= r396064;
        double r396066 = 63485.56364381746;
        bool r396067 = r396063 <= r396066;
        double r396068 = !r396067;
        bool r396069 = r396065 || r396068;
        double r396070 = x;
        double r396071 = 0.0692910599291889;
        double r396072 = y;
        double r396073 = r396071 * r396072;
        double r396074 = r396072 / r396063;
        double r396075 = 0.07512208616047561;
        double r396076 = 0.40462203869992125;
        double r396077 = r396076 / r396063;
        double r396078 = r396075 - r396077;
        double r396079 = r396074 * r396078;
        double r396080 = r396073 + r396079;
        double r396081 = r396070 + r396080;
        double r396082 = r396063 * r396071;
        double r396083 = 0.4917317610505968;
        double r396084 = r396082 + r396083;
        double r396085 = cbrt(r396084);
        double r396086 = r396085 * r396085;
        double r396087 = r396085 * r396063;
        double r396088 = r396086 * r396087;
        double r396089 = 0.279195317918525;
        double r396090 = r396088 + r396089;
        double r396091 = 6.012459259764103;
        double r396092 = r396063 + r396091;
        double r396093 = r396092 * r396063;
        double r396094 = 3.350343815022304;
        double r396095 = r396093 + r396094;
        double r396096 = r396090 / r396095;
        double r396097 = r396072 * r396096;
        double r396098 = r396070 + r396097;
        double r396099 = r396069 ? r396081 : r396098;
        return r396099;
}

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

Original19.3
Target0.2
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 < -2549507.4500260064 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-identity39.5

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

    4. Applied times-frac31.5

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

    5. Simplified31.5

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -2549507.4500260064 < 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-identity0.2

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

    4. Applied times-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-cbrt0.1

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

    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 simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2549507.4500260064 \lor \neg \left(z \le 63485.5636438174624\right):\\ \;\;\;\;x + \left(0.0692910599291888946 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.404622038699921249}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right) + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

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))))