Average Error: 20.4 → 0.1
Time: 3.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.44766332686604399 \cdot 10^{27} \lor \neg \left(z \le 206714745.849681407\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, 6.0124592597641033, \mathsf{fma}\left(z, z, 3.35034381502230394\right)\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)\\ \end{array}\]
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 -6.44766332686604399 \cdot 10^{27} \lor \neg \left(z \le 206714745.849681407\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r350017 = x;
        double r350018 = y;
        double r350019 = z;
        double r350020 = 0.0692910599291889;
        double r350021 = r350019 * r350020;
        double r350022 = 0.4917317610505968;
        double r350023 = r350021 + r350022;
        double r350024 = r350023 * r350019;
        double r350025 = 0.279195317918525;
        double r350026 = r350024 + r350025;
        double r350027 = r350018 * r350026;
        double r350028 = 6.012459259764103;
        double r350029 = r350019 + r350028;
        double r350030 = r350029 * r350019;
        double r350031 = 3.350343815022304;
        double r350032 = r350030 + r350031;
        double r350033 = r350027 / r350032;
        double r350034 = r350017 + r350033;
        return r350034;
}
↓
double f(double x, double y, double z) {
        double r350035 = z;
        double r350036 = -6.447663326866044e+27;
        bool r350037 = r350035 <= r350036;
        double r350038 = 206714745.8496814;
        bool r350039 = r350035 <= r350038;
        double r350040 = !r350039;
        bool r350041 = r350037 || r350040;
        double r350042 = 0.07512208616047561;
        double r350043 = r350042 / r350035;
        double r350044 = y;
        double r350045 = 0.0692910599291889;
        double r350046 = x;
        double r350047 = fma(r350044, r350045, r350046);
        double r350048 = fma(r350043, r350044, r350047);
        double r350049 = 6.012459259764103;
        double r350050 = 3.350343815022304;
        double r350051 = fma(r350035, r350035, r350050);
        double r350052 = fma(r350035, r350049, r350051);
        double r350053 = r350044 / r350052;
        double r350054 = 0.4917317610505968;
        double r350055 = fma(r350035, r350045, r350054);
        double r350056 = 0.279195317918525;
        double r350057 = fma(r350055, r350035, r350056);
        double r350058 = fma(r350053, r350057, r350046);
        double r350059 = r350041 ? r350048 : r350058;
        return r350059;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.4
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 < -6.447663326866044e+27 or 206714745.8496814 < z

    1. Initial program 42.8

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

    2. Simplified35.2

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

    3. 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 -6.447663326866044e+27 < z < 206714745.8496814

    1. Initial program 0.3

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

    2. Simplified0.1

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

    3. Taylor expanded around 0 0.1

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{6.0124592597641033 \cdot z + \left({z}^{2} + 3.35034381502230394\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)\]

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.44766332686604399 \cdot 10^{27} \lor \neg \left(z \le 206714745.849681407\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, 6.0124592597641033, \mathsf{fma}\left(z, z, 3.35034381502230394\right)\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))))