Average Error: 19.9 → 0.2
Time: 19.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2425787276974239638204973056:\\ \;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \mathbf{elif}\;z \le 124304173.40660762786865234375:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736\right)}{3.350343815022303939343828460550867021084 + z \cdot \left(z + 6.012459259764103336465268512256443500519\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2425787276974239638204973056:\\
\;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\

\mathbf{elif}\;z \le 124304173.40660762786865234375:\\
\;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736\right)}{3.350343815022303939343828460550867021084 + z \cdot \left(z + 6.012459259764103336465268512256443500519\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r15254075 = x;
        double r15254076 = y;
        double r15254077 = z;
        double r15254078 = 0.0692910599291889;
        double r15254079 = r15254077 * r15254078;
        double r15254080 = 0.4917317610505968;
        double r15254081 = r15254079 + r15254080;
        double r15254082 = r15254081 * r15254077;
        double r15254083 = 0.279195317918525;
        double r15254084 = r15254082 + r15254083;
        double r15254085 = r15254076 * r15254084;
        double r15254086 = 6.012459259764103;
        double r15254087 = r15254077 + r15254086;
        double r15254088 = r15254087 * r15254077;
        double r15254089 = 3.350343815022304;
        double r15254090 = r15254088 + r15254089;
        double r15254091 = r15254085 / r15254090;
        double r15254092 = r15254075 + r15254091;
        return r15254092;
}
↓
double f(double x, double y, double z) {
        double r15254093 = z;
        double r15254094 = -2.4257872769742396e+27;
        bool r15254095 = r15254093 <= r15254094;
        double r15254096 = y;
        double r15254097 = 0.0692910599291889;
        double r15254098 = r15254096 / r15254093;
        double r15254099 = 0.07512208616047561;
        double r15254100 = x;
        double r15254101 = fma(r15254098, r15254099, r15254100);
        double r15254102 = fma(r15254096, r15254097, r15254101);
        double r15254103 = 124304173.40660763;
        bool r15254104 = r15254093 <= r15254103;
        double r15254105 = r15254097 * r15254093;
        double r15254106 = 0.4917317610505968;
        double r15254107 = r15254105 + r15254106;
        double r15254108 = r15254093 * r15254107;
        double r15254109 = 0.279195317918525;
        double r15254110 = r15254108 + r15254109;
        double r15254111 = r15254096 * r15254110;
        double r15254112 = 3.350343815022304;
        double r15254113 = 6.012459259764103;
        double r15254114 = r15254093 + r15254113;
        double r15254115 = r15254093 * r15254114;
        double r15254116 = r15254112 + r15254115;
        double r15254117 = r15254111 / r15254116;
        double r15254118 = r15254100 + r15254117;
        double r15254119 = r15254104 ? r15254118 : r15254102;
        double r15254120 = r15254095 ? r15254102 : r15254119;
        return r15254120;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.9
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.4257872769742396e+27 or 124304173.40660763 < z

    1. Initial program 41.9

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

    2. Simplified35.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)}\]

    3. Taylor expanded around 0 35.6

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{6.012459259764103336465268512256443500519 \cdot z + \left({z}^{2} + 3.350343815022303939343828460550867021084\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]

    4. Simplified35.6

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103336465268512256443500519, 3.350343815022303939343828460550867021084\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]

    5. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x + \left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right)}\]

    6. Simplified0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)}\]

    if -2.4257872769742396e+27 < z < 124304173.40660763

    1. Initial program 0.3

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2425787276974239638204973056:\\ \;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \mathbf{elif}\;z \le 124304173.40660762786865234375:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736\right)}{3.350343815022303939343828460550867021084 + z \cdot \left(z + 6.012459259764103336465268512256443500519\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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)) (/ 1.0 (+ (* (+ 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))))