Average Error: 19.3 → 0.1
Time: 10.3s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -457625405627486460 \lor \neg \left(z \le 63485.5636438174624\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291888946, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)\\ \end{array}\]
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 -457625405627486460 \lor \neg \left(z \le 63485.5636438174624\right):\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291888946, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x, double y, double z) {
        double r373972 = x;
        double r373973 = y;
        double r373974 = z;
        double r373975 = 0.0692910599291889;
        double r373976 = r373974 * r373975;
        double r373977 = 0.4917317610505968;
        double r373978 = r373976 + r373977;
        double r373979 = r373978 * r373974;
        double r373980 = 0.279195317918525;
        double r373981 = r373979 + r373980;
        double r373982 = r373973 * r373981;
        double r373983 = 6.012459259764103;
        double r373984 = r373974 + r373983;
        double r373985 = r373984 * r373974;
        double r373986 = 3.350343815022304;
        double r373987 = r373985 + r373986;
        double r373988 = r373982 / r373987;
        double r373989 = r373972 + r373988;
        return r373989;
}
↓
double f(double x, double y, double z) {
        double r373990 = z;
        double r373991 = -4.5762540562748646e+17;
        bool r373992 = r373990 <= r373991;
        double r373993 = 63485.56364381746;
        bool r373994 = r373990 <= r373993;
        double r373995 = !r373994;
        bool r373996 = r373992 || r373995;
        double r373997 = 0.0692910599291889;
        double r373998 = y;
        double r373999 = 0.07512208616047561;
        double r374000 = r373998 / r373990;
        double r374001 = x;
        double r374002 = fma(r373999, r374000, r374001);
        double r374003 = fma(r373997, r373998, r374002);
        double r374004 = 6.012459259764103;
        double r374005 = r373990 + r374004;
        double r374006 = 3.350343815022304;
        double r374007 = fma(r374005, r373990, r374006);
        double r374008 = r373998 / r374007;
        double r374009 = 0.4917317610505968;
        double r374010 = fma(r373990, r373997, r374009);
        double r374011 = 0.279195317918525;
        double r374012 = fma(r374010, r373990, r374011);
        double r374013 = fma(r374008, r374012, r374001);
        double r374014 = r373996 ? r374003 : r374013;
        return r374014;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.3
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 < -4.5762540562748646e+17 or 63485.56364381746 < z

    1. Initial program 40.2

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

    2. Simplified33.2

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

    4. Applied add-cube-cbrt33.4

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

    5. Taylor expanded around inf 0.0

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

    6. Simplified0.0

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

    if -4.5762540562748646e+17 < z < 63485.56364381746

    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. 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. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -457625405627486460 \lor \neg \left(z \le 63485.5636438174624\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291888946, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))