Average Error: 20.0 → 0.2
Time: 17.3s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -123702675580259303424 \lor \neg \left(z \le 0.02992000358306671353725292306080518756062\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}, y, x\right)\\ \end{array}\]
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 -123702675580259303424 \lor \neg \left(z \le 0.02992000358306671353725292306080518756062\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}, y, x\right)\\

\end{array}
double f(double x, double y, double z) {
        double r235153 = x;
        double r235154 = y;
        double r235155 = z;
        double r235156 = 0.0692910599291889;
        double r235157 = r235155 * r235156;
        double r235158 = 0.4917317610505968;
        double r235159 = r235157 + r235158;
        double r235160 = r235159 * r235155;
        double r235161 = 0.279195317918525;
        double r235162 = r235160 + r235161;
        double r235163 = r235154 * r235162;
        double r235164 = 6.012459259764103;
        double r235165 = r235155 + r235164;
        double r235166 = r235165 * r235155;
        double r235167 = 3.350343815022304;
        double r235168 = r235166 + r235167;
        double r235169 = r235163 / r235168;
        double r235170 = r235153 + r235169;
        return r235170;
}
↓
double f(double x, double y, double z) {
        double r235171 = z;
        double r235172 = -1.237026755802593e+20;
        bool r235173 = r235171 <= r235172;
        double r235174 = 0.029920003583066714;
        bool r235175 = r235171 <= r235174;
        double r235176 = !r235175;
        bool r235177 = r235173 || r235176;
        double r235178 = y;
        double r235179 = r235178 / r235171;
        double r235180 = 0.07512208616047561;
        double r235181 = 0.0692910599291889;
        double r235182 = x;
        double r235183 = fma(r235178, r235181, r235182);
        double r235184 = fma(r235179, r235180, r235183);
        double r235185 = 0.4917317610505968;
        double r235186 = fma(r235181, r235171, r235185);
        double r235187 = 0.279195317918525;
        double r235188 = fma(r235186, r235171, r235187);
        double r235189 = 6.012459259764103;
        double r235190 = r235189 + r235171;
        double r235191 = 3.350343815022304;
        double r235192 = fma(r235190, r235171, r235191);
        double r235193 = r235188 / r235192;
        double r235194 = fma(r235193, r235178, r235182);
        double r235195 = r235177 ? r235184 : r235194;
        return r235195;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.0
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 < -1.237026755802593e+20 or 0.029920003583066714 < z

    1. Initial program 41.4

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    if -1.237026755802593e+20 < z < 0.029920003583066714

    1. Initial program 0.2

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

    2. Simplified0.1

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

    3. Taylor expanded around 0 0.1

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

    4. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -123702675580259303424 \lor \neg \left(z \le 0.02992000358306671353725292306080518756062\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))))