Average Error: 19.7 → 0.1
Time: 4.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -96048976883392.062 \lor \neg \left(z \le 680550425.79129004\right):\\ \;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1 \cdot \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 -96048976883392.062 \lor \neg \left(z \le 680550425.79129004\right):\\
\;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\

\mathbf{else}:\\
\;\;\;\;x + 1 \cdot \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}\\

\end{array}
double f(double x, double y, double z) {
        double r449860 = x;
        double r449861 = y;
        double r449862 = z;
        double r449863 = 0.0692910599291889;
        double r449864 = r449862 * r449863;
        double r449865 = 0.4917317610505968;
        double r449866 = r449864 + r449865;
        double r449867 = r449866 * r449862;
        double r449868 = 0.279195317918525;
        double r449869 = r449867 + r449868;
        double r449870 = r449861 * r449869;
        double r449871 = 6.012459259764103;
        double r449872 = r449862 + r449871;
        double r449873 = r449872 * r449862;
        double r449874 = 3.350343815022304;
        double r449875 = r449873 + r449874;
        double r449876 = r449870 / r449875;
        double r449877 = r449860 + r449876;
        return r449877;
}
↓
double f(double x, double y, double z) {
        double r449878 = z;
        double r449879 = -96048976883392.06;
        bool r449880 = r449878 <= r449879;
        double r449881 = 680550425.79129;
        bool r449882 = r449878 <= r449881;
        double r449883 = !r449882;
        bool r449884 = r449880 || r449883;
        double r449885 = x;
        double r449886 = 0.07512208616047561;
        double r449887 = y;
        double r449888 = r449887 / r449878;
        double r449889 = r449886 * r449888;
        double r449890 = 0.0692910599291889;
        double r449891 = r449890 * r449887;
        double r449892 = r449889 + r449891;
        double r449893 = r449885 + r449892;
        double r449894 = 1.0;
        double r449895 = r449878 * r449890;
        double r449896 = 0.4917317610505968;
        double r449897 = r449895 + r449896;
        double r449898 = r449897 * r449878;
        double r449899 = 0.279195317918525;
        double r449900 = r449898 + r449899;
        double r449901 = r449887 * r449900;
        double r449902 = 6.012459259764103;
        double r449903 = r449878 + r449902;
        double r449904 = r449903 * r449878;
        double r449905 = 3.350343815022304;
        double r449906 = r449904 + r449905;
        double r449907 = r449901 / r449906;
        double r449908 = r449894 * r449907;
        double r449909 = r449885 + r449908;
        double r449910 = r449884 ? r449893 : r449909;
        return r449910;
}

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

Original19.7
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 < -96048976883392.06 or 680550425.79129 < z

    1. Initial program 41.0

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

    2. Taylor expanded around inf 0.0

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

    if -96048976883392.06 < z < 680550425.79129

    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 add-sqr-sqrt0.6

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

    4. Applied times-frac0.3

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

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

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

    7. Applied associate-*l*0.3

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

    8. Simplified0.2

      \[\leadsto x + 1 \cdot \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -96048976883392.062 \lor \neg \left(z \le 680550425.79129004\right):\\ \;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1 \cdot \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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))))