Average Error: 20.5 → 0.1
Time: 3.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.03846987469645444 \cdot 10^{39} \lor \neg \left(z \le 195069351.271798879\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\frac{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}{y}} + x\\ \end{array}\]
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 -2.03846987469645444 \cdot 10^{39} \lor \neg \left(z \le 195069351.271798879\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\frac{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}{y}} + x\\

\end{array}
double f(double x, double y, double z) {
        double r313863 = x;
        double r313864 = y;
        double r313865 = z;
        double r313866 = 0.0692910599291889;
        double r313867 = r313865 * r313866;
        double r313868 = 0.4917317610505968;
        double r313869 = r313867 + r313868;
        double r313870 = r313869 * r313865;
        double r313871 = 0.279195317918525;
        double r313872 = r313870 + r313871;
        double r313873 = r313864 * r313872;
        double r313874 = 6.012459259764103;
        double r313875 = r313865 + r313874;
        double r313876 = r313875 * r313865;
        double r313877 = 3.350343815022304;
        double r313878 = r313876 + r313877;
        double r313879 = r313873 / r313878;
        double r313880 = r313863 + r313879;
        return r313880;
}
↓
double f(double x, double y, double z) {
        double r313881 = z;
        double r313882 = -2.0384698746964544e+39;
        bool r313883 = r313881 <= r313882;
        double r313884 = 195069351.27179888;
        bool r313885 = r313881 <= r313884;
        double r313886 = !r313885;
        bool r313887 = r313883 || r313886;
        double r313888 = 0.07512208616047561;
        double r313889 = r313888 / r313881;
        double r313890 = y;
        double r313891 = 0.0692910599291889;
        double r313892 = x;
        double r313893 = fma(r313890, r313891, r313892);
        double r313894 = fma(r313889, r313890, r313893);
        double r313895 = 0.4917317610505968;
        double r313896 = fma(r313881, r313891, r313895);
        double r313897 = 0.279195317918525;
        double r313898 = fma(r313896, r313881, r313897);
        double r313899 = 6.012459259764103;
        double r313900 = r313881 + r313899;
        double r313901 = 3.350343815022304;
        double r313902 = fma(r313900, r313881, r313901);
        double r313903 = r313902 / r313890;
        double r313904 = r313898 / r313903;
        double r313905 = r313904 + r313892;
        double r313906 = r313887 ? r313894 : r313905;
        return r313906;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.5
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 < -2.0384698746964544e+39 or 195069351.27179888 < z

    1. Initial program 43.7

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

    2. Simplified35.7

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)}\]

    if -2.0384698746964544e+39 < z < 195069351.27179888

    1. Initial program 0.5

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

    2. Simplified0.2

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

    4. Applied add-cube-cbrt0.5

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

    5. Applied associate-/r*0.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{\sqrt[3]{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}}}{\sqrt[3]{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)\]
    6. Using strategy rm

    7. Applied fma-udef0.3

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

    8. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.03846987469645444 \cdot 10^{39} \lor \neg \left(z \le 195069351.271798879\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\frac{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}{y}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(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))))