Average Error: 19.3 → 0.1
Time: 18.4s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -676045330612.146:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{elif}\;z \le 95815542.39360864:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, 6.012459259764103 + z, 3.350343815022304\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}
↓
\begin{array}{l}
\mathbf{if}\;z \le -676045330612.146:\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\

\mathbf{elif}\;z \le 95815542.39360864:\\
\;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, 6.012459259764103 + z, 3.350343815022304\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r13945918 = x;
        double r13945919 = y;
        double r13945920 = z;
        double r13945921 = 0.0692910599291889;
        double r13945922 = r13945920 * r13945921;
        double r13945923 = 0.4917317610505968;
        double r13945924 = r13945922 + r13945923;
        double r13945925 = r13945924 * r13945920;
        double r13945926 = 0.279195317918525;
        double r13945927 = r13945925 + r13945926;
        double r13945928 = r13945919 * r13945927;
        double r13945929 = 6.012459259764103;
        double r13945930 = r13945920 + r13945929;
        double r13945931 = r13945930 * r13945920;
        double r13945932 = 3.350343815022304;
        double r13945933 = r13945931 + r13945932;
        double r13945934 = r13945928 / r13945933;
        double r13945935 = r13945918 + r13945934;
        return r13945935;
}
↓
double f(double x, double y, double z) {
        double r13945936 = z;
        double r13945937 = -676045330612.146;
        bool r13945938 = r13945936 <= r13945937;
        double r13945939 = 0.0692910599291889;
        double r13945940 = y;
        double r13945941 = 0.07512208616047561;
        double r13945942 = r13945940 / r13945936;
        double r13945943 = x;
        double r13945944 = fma(r13945941, r13945942, r13945943);
        double r13945945 = fma(r13945939, r13945940, r13945944);
        double r13945946 = 95815542.39360864;
        bool r13945947 = r13945936 <= r13945946;
        double r13945948 = 0.4917317610505968;
        double r13945949 = fma(r13945936, r13945939, r13945948);
        double r13945950 = 0.279195317918525;
        double r13945951 = fma(r13945936, r13945949, r13945950);
        double r13945952 = r13945940 * r13945951;
        double r13945953 = 6.012459259764103;
        double r13945954 = r13945953 + r13945936;
        double r13945955 = 3.350343815022304;
        double r13945956 = fma(r13945936, r13945954, r13945955);
        double r13945957 = r13945952 / r13945956;
        double r13945958 = r13945943 + r13945957;
        double r13945959 = r13945947 ? r13945958 : r13945945;
        double r13945960 = r13945938 ? r13945945 : r13945959;
        return r13945960;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.3
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.652456675:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -676045330612.146 or 95815542.39360864 < z

    1. Initial program 39.8

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

    2. Simplified34.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}, \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), x\right)}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291889 \cdot y\right)}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)}\]

    if -676045330612.146 < z < 95815542.39360864

    1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

    2. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}, \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), x\right)}\]
    3. Using strategy rm

    4. Applied clear-num0.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}{y}}}, \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), x\right)\]
    5. Using strategy rm

    6. Applied div-inv0.2

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right) \cdot \frac{1}{y}}}, \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), x\right)\]

    7. Applied add-cube-cbrt0.2

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right) \cdot \frac{1}{y}}, \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), x\right)\]

    8. Applied times-frac0.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)} \cdot \frac{\sqrt[3]{1}}{\frac{1}{y}}}, \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), x\right)\]

    9. Simplified0.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{y}}, \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), x\right)\]

    10. Simplified0.2

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)} \cdot \color{blue}{y}, \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), x\right)\]
    11. Using strategy rm

    12. Applied fma-udef0.2

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)} \cdot y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right) + x}\]

    13. Simplified0.2

      \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -676045330612.146:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{elif}\;z \le 95815542.39360864:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, 6.012459259764103 + z, 3.350343815022304\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(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 (+ (* (+ 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))))