Average Error: 20.6 → 0.1
Time: 21.2s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -464704203.059111535549163818359375:\\ \;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \mathbf{elif}\;z \le 46621187.154576636850833892822265625:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z}\right) \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z} + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \end{array}\]
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 -464704203.059111535549163818359375:\\
\;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\

\mathbf{elif}\;z \le 46621187.154576636850833892822265625:\\
\;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z}\right) \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z} + 3.350343815022303939343828460550867021084}\\

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

\end{array}
double f(double x, double y, double z) {
        double r23988092 = x;
        double r23988093 = y;
        double r23988094 = z;
        double r23988095 = 0.0692910599291889;
        double r23988096 = r23988094 * r23988095;
        double r23988097 = 0.4917317610505968;
        double r23988098 = r23988096 + r23988097;
        double r23988099 = r23988098 * r23988094;
        double r23988100 = 0.279195317918525;
        double r23988101 = r23988099 + r23988100;
        double r23988102 = r23988093 * r23988101;
        double r23988103 = 6.012459259764103;
        double r23988104 = r23988094 + r23988103;
        double r23988105 = r23988104 * r23988094;
        double r23988106 = 3.350343815022304;
        double r23988107 = r23988105 + r23988106;
        double r23988108 = r23988102 / r23988107;
        double r23988109 = r23988092 + r23988108;
        return r23988109;
}
↓
double f(double x, double y, double z) {
        double r23988110 = z;
        double r23988111 = -464704203.05911154;
        bool r23988112 = r23988110 <= r23988111;
        double r23988113 = 0.0692910599291889;
        double r23988114 = y;
        double r23988115 = r23988114 / r23988110;
        double r23988116 = 0.07512208616047561;
        double r23988117 = x;
        double r23988118 = fma(r23988115, r23988116, r23988117);
        double r23988119 = fma(r23988113, r23988114, r23988118);
        double r23988120 = 46621187.15457664;
        bool r23988121 = r23988110 <= r23988120;
        double r23988122 = r23988110 * r23988113;
        double r23988123 = 0.4917317610505968;
        double r23988124 = r23988122 + r23988123;
        double r23988125 = r23988124 * r23988110;
        double r23988126 = 0.279195317918525;
        double r23988127 = r23988125 + r23988126;
        double r23988128 = r23988114 * r23988127;
        double r23988129 = 6.012459259764103;
        double r23988130 = r23988110 + r23988129;
        double r23988131 = r23988130 * r23988110;
        double r23988132 = cbrt(r23988131);
        double r23988133 = r23988132 * r23988132;
        double r23988134 = r23988133 * r23988132;
        double r23988135 = 3.350343815022304;
        double r23988136 = r23988134 + r23988135;
        double r23988137 = r23988128 / r23988136;
        double r23988138 = r23988117 + r23988137;
        double r23988139 = r23988121 ? r23988138 : r23988119;
        double r23988140 = r23988112 ? r23988119 : r23988139;
        return r23988140;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.6
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -464704203.05911154 or 46621187.15457664 < z

    1. Initial program 41.8

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

    2. Simplified34.2

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -464704203.05911154 < z < 46621187.15457664

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -464704203.059111535549163818359375:\\ \;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \mathbf{elif}\;z \le 46621187.154576636850833892822265625:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z}\right) \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z} + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))))