Average Error: 19.8 → 0.1
Time: 4.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.1817420192949543 \cdot 10^{39} \lor \neg \left(z \le 1326132.872118698\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\\ \end{array}\]
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 -3.1817420192949543 \cdot 10^{39} \lor \neg \left(z \le 1326132.872118698\right):\\
\;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r435908 = x;
        double r435909 = y;
        double r435910 = z;
        double r435911 = 0.0692910599291889;
        double r435912 = r435910 * r435911;
        double r435913 = 0.4917317610505968;
        double r435914 = r435912 + r435913;
        double r435915 = r435914 * r435910;
        double r435916 = 0.279195317918525;
        double r435917 = r435915 + r435916;
        double r435918 = r435909 * r435917;
        double r435919 = 6.012459259764103;
        double r435920 = r435910 + r435919;
        double r435921 = r435920 * r435910;
        double r435922 = 3.350343815022304;
        double r435923 = r435921 + r435922;
        double r435924 = r435918 / r435923;
        double r435925 = r435908 + r435924;
        return r435925;
}
↓
double f(double x, double y, double z) {
        double r435926 = z;
        double r435927 = -3.1817420192949543e+39;
        bool r435928 = r435926 <= r435927;
        double r435929 = 1326132.872118698;
        bool r435930 = r435926 <= r435929;
        double r435931 = !r435930;
        bool r435932 = r435928 || r435931;
        double r435933 = x;
        double r435934 = 0.07512208616047561;
        double r435935 = y;
        double r435936 = r435935 / r435926;
        double r435937 = r435934 * r435936;
        double r435938 = 0.0692910599291889;
        double r435939 = r435938 * r435935;
        double r435940 = r435937 + r435939;
        double r435941 = 0.40462203869992125;
        double r435942 = 2.0;
        double r435943 = pow(r435926, r435942);
        double r435944 = r435935 / r435943;
        double r435945 = r435941 * r435944;
        double r435946 = r435940 - r435945;
        double r435947 = r435933 + r435946;
        double r435948 = 6.012459259764103;
        double r435949 = r435926 + r435948;
        double r435950 = r435949 * r435926;
        double r435951 = 3.350343815022304;
        double r435952 = r435950 + r435951;
        double r435953 = sqrt(r435952);
        double r435954 = r435935 / r435953;
        double r435955 = r435926 * r435938;
        double r435956 = 0.4917317610505968;
        double r435957 = r435955 + r435956;
        double r435958 = r435957 * r435926;
        double r435959 = 0.279195317918525;
        double r435960 = r435958 + r435959;
        double r435961 = r435960 / r435953;
        double r435962 = r435954 * r435961;
        double r435963 = r435933 + r435962;
        double r435964 = r435932 ? r435947 : r435963;
        return r435964;
}

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

Original19.8
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 < -3.1817420192949543e+39 or 1326132.872118698 < z

    1. Initial program 42.9

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

    2. Taylor expanded around inf 0.0

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

    if -3.1817420192949543e+39 < z < 1326132.872118698

    1. Initial program 0.4

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

    3. Applied add-sqr-sqrt0.7

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

    4. Applied times-frac0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.1817420192949543 \cdot 10^{39} \lor \neg \left(z \le 1326132.872118698\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))))