Average Error: 20.1 → 0.1
Time: 20.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -23788968548377700:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\ \mathbf{elif}\;z \le 453539.7170896522584371268749237060546875:\\ \;\;\;\;\frac{y \cdot \left(0.2791953179185249767080279070796677842736 + \left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z\right)}{3.350343815022303939343828460550867021084 + \left(z + 6.012459259764103336465268512256443500519\right) \cdot z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\ \end{array}\]
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 -23788968548377700:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\

\mathbf{elif}\;z \le 453539.7170896522584371268749237060546875:\\
\;\;\;\;\frac{y \cdot \left(0.2791953179185249767080279070796677842736 + \left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z\right)}{3.350343815022303939343828460550867021084 + \left(z + 6.012459259764103336465268512256443500519\right) \cdot z} + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\

\end{array}
double f(double x, double y, double z) {
        double r13015160 = x;
        double r13015161 = y;
        double r13015162 = z;
        double r13015163 = 0.0692910599291889;
        double r13015164 = r13015162 * r13015163;
        double r13015165 = 0.4917317610505968;
        double r13015166 = r13015164 + r13015165;
        double r13015167 = r13015166 * r13015162;
        double r13015168 = 0.279195317918525;
        double r13015169 = r13015167 + r13015168;
        double r13015170 = r13015161 * r13015169;
        double r13015171 = 6.012459259764103;
        double r13015172 = r13015162 + r13015171;
        double r13015173 = r13015172 * r13015162;
        double r13015174 = 3.350343815022304;
        double r13015175 = r13015173 + r13015174;
        double r13015176 = r13015170 / r13015175;
        double r13015177 = r13015160 + r13015176;
        return r13015177;
}
↓
double f(double x, double y, double z) {
        double r13015178 = z;
        double r13015179 = -2.37889685483777e+16;
        bool r13015180 = r13015178 <= r13015179;
        double r13015181 = 1.0;
        double r13015182 = 101.23733352003816;
        double r13015183 = r13015178 * r13015178;
        double r13015184 = r13015182 / r13015183;
        double r13015185 = 15.646356830292035;
        double r13015186 = r13015185 / r13015178;
        double r13015187 = r13015184 - r13015186;
        double r13015188 = 14.431876219268938;
        double r13015189 = r13015187 + r13015188;
        double r13015190 = r13015181 / r13015189;
        double r13015191 = y;
        double r13015192 = x;
        double r13015193 = fma(r13015190, r13015191, r13015192);
        double r13015194 = 453539.71708965226;
        bool r13015195 = r13015178 <= r13015194;
        double r13015196 = 0.279195317918525;
        double r13015197 = 0.4917317610505968;
        double r13015198 = 0.0692910599291889;
        double r13015199 = r13015178 * r13015198;
        double r13015200 = r13015197 + r13015199;
        double r13015201 = r13015200 * r13015178;
        double r13015202 = r13015196 + r13015201;
        double r13015203 = r13015191 * r13015202;
        double r13015204 = 3.350343815022304;
        double r13015205 = 6.012459259764103;
        double r13015206 = r13015178 + r13015205;
        double r13015207 = r13015206 * r13015178;
        double r13015208 = r13015204 + r13015207;
        double r13015209 = r13015203 / r13015208;
        double r13015210 = r13015209 + r13015192;
        double r13015211 = r13015195 ? r13015210 : r13015193;
        double r13015212 = r13015180 ? r13015193 : r13015211;
        return r13015212;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.1
Target0.1
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 < -2.37889685483777e+16 or 453539.71708965226 < z

    1. Initial program 41.5

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

    2. Simplified33.4

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

    4. Applied clear-num33.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}}, y, x\right)\]

    5. Taylor expanded around inf 0.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\left(101.237333520038163214849191717803478241 \cdot \frac{1}{{z}^{2}} + 14.43187621926893804413793986896052956581\right) - 15.64635683029203505611803848296403884888 \cdot \frac{1}{z}}}, y, x\right)\]

    6. Simplified0.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{14.43187621926893804413793986896052956581 + \left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right)}}, y, x\right)\]

    if -2.37889685483777e+16 < z < 453539.71708965226

    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 simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -23788968548377700:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\ \mathbf{elif}\;z \le 453539.7170896522584371268749237060546875:\\ \;\;\;\;\frac{y \cdot \left(0.2791953179185249767080279070796677842736 + \left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z\right)}{3.350343815022303939343828460550867021084 + \left(z + 6.012459259764103336465268512256443500519\right) \cdot z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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))))