Average Error: 20.0 → 0.1
Time: 8.1s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.36903654096377077 \cdot 10^{69} \lor \neg \left(z \le 2180725.6674509291\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291888946, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}\\ \end{array}\]
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 -3.36903654096377077 \cdot 10^{69} \lor \neg \left(z \le 2180725.6674509291\right):\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291888946, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r320031 = x;
        double r320032 = y;
        double r320033 = z;
        double r320034 = 0.0692910599291889;
        double r320035 = r320033 * r320034;
        double r320036 = 0.4917317610505968;
        double r320037 = r320035 + r320036;
        double r320038 = r320037 * r320033;
        double r320039 = 0.279195317918525;
        double r320040 = r320038 + r320039;
        double r320041 = r320032 * r320040;
        double r320042 = 6.012459259764103;
        double r320043 = r320033 + r320042;
        double r320044 = r320043 * r320033;
        double r320045 = 3.350343815022304;
        double r320046 = r320044 + r320045;
        double r320047 = r320041 / r320046;
        double r320048 = r320031 + r320047;
        return r320048;
}
↓
double f(double x, double y, double z) {
        double r320049 = z;
        double r320050 = -3.3690365409637708e+69;
        bool r320051 = r320049 <= r320050;
        double r320052 = 2180725.667450929;
        bool r320053 = r320049 <= r320052;
        double r320054 = !r320053;
        bool r320055 = r320051 || r320054;
        double r320056 = 0.0692910599291889;
        double r320057 = y;
        double r320058 = 0.07512208616047561;
        double r320059 = r320057 / r320049;
        double r320060 = x;
        double r320061 = fma(r320058, r320059, r320060);
        double r320062 = fma(r320056, r320057, r320061);
        double r320063 = 0.4917317610505968;
        double r320064 = fma(r320049, r320056, r320063);
        double r320065 = 0.279195317918525;
        double r320066 = fma(r320064, r320049, r320065);
        double r320067 = 6.012459259764103;
        double r320068 = r320049 + r320067;
        double r320069 = 3.350343815022304;
        double r320070 = fma(r320068, r320049, r320069);
        double r320071 = r320066 / r320070;
        double r320072 = r320057 * r320071;
        double r320073 = r320060 + r320072;
        double r320074 = r320055 ? r320062 : r320073;
        return r320074;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.0
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 < -3.3690365409637708e+69 or 2180725.667450929 < z

    1. Initial program 45.1

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

    2. Simplified38.3

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

    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

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

    if -3.3690365409637708e+69 < z < 2180725.667450929

    1. Initial program 0.8

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

    3. Applied *-un-lft-identity0.8

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\color{blue}{1 \cdot \left(\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394\right)}}\]

    4. Applied times-frac0.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\]

    5. Simplified0.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.36903654096377077 \cdot 10^{69} \lor \neg \left(z \le 2180725.6674509291\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291888946, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))