Average Error: 20.4 → 0.3
Time: 14.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -252224470884185145344 \lor \neg \left(z \le 1.611737647351515965898536252021155945613 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]
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 -252224470884185145344 \lor \neg \left(z \le 1.611737647351515965898536252021155945613 \cdot 10^{-6}\right):\\
\;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r230090 = x;
        double r230091 = y;
        double r230092 = z;
        double r230093 = 0.0692910599291889;
        double r230094 = r230092 * r230093;
        double r230095 = 0.4917317610505968;
        double r230096 = r230094 + r230095;
        double r230097 = r230096 * r230092;
        double r230098 = 0.279195317918525;
        double r230099 = r230097 + r230098;
        double r230100 = r230091 * r230099;
        double r230101 = 6.012459259764103;
        double r230102 = r230092 + r230101;
        double r230103 = r230102 * r230092;
        double r230104 = 3.350343815022304;
        double r230105 = r230103 + r230104;
        double r230106 = r230100 / r230105;
        double r230107 = r230090 + r230106;
        return r230107;
}
↓
double f(double x, double y, double z) {
        double r230108 = z;
        double r230109 = -2.5222447088418515e+20;
        bool r230110 = r230108 <= r230109;
        double r230111 = 1.611737647351516e-06;
        bool r230112 = r230108 <= r230111;
        double r230113 = !r230112;
        bool r230114 = r230110 || r230113;
        double r230115 = 0.07512208616047561;
        double r230116 = y;
        double r230117 = r230116 / r230108;
        double r230118 = 0.0692910599291889;
        double r230119 = x;
        double r230120 = fma(r230116, r230118, r230119);
        double r230121 = fma(r230115, r230117, r230120);
        double r230122 = r230108 * r230118;
        double r230123 = 0.4917317610505968;
        double r230124 = r230122 + r230123;
        double r230125 = r230124 * r230108;
        double r230126 = 0.279195317918525;
        double r230127 = r230125 + r230126;
        double r230128 = r230116 * r230127;
        double r230129 = 6.012459259764103;
        double r230130 = r230108 + r230129;
        double r230131 = r230130 * r230108;
        double r230132 = 3.350343815022304;
        double r230133 = r230131 + r230132;
        double r230134 = r230128 / r230133;
        double r230135 = r230119 + r230134;
        double r230136 = r230114 ? r230121 : r230135;
        return r230136;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.4
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.5222447088418515e+20 or 1.611737647351516e-06 < z

    1. Initial program 41.7

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

    2. Simplified34.5

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

    3. Taylor expanded around inf 0.4

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

    4. Simplified0.4

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

    if -2.5222447088418515e+20 < z < 1.611737647351516e-06

    1. Initial program 0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -252224470884185145344 \lor \neg \left(z \le 1.611737647351515965898536252021155945613 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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 -8120153.6524566747) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (/ 1 (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394))))