Average Error: 20.1 → 0.2
Time: 1.1m
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4230301041346655513100977098981376:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \mathbf{elif}\;z \le 203544506.3211925327777862548828125:\\ \;\;\;\;x + \frac{y}{\sqrt{3.350343815022303939343828460550867021084 + \left(6.012459259764103336465268512256443500519 + z\right) \cdot z}} \cdot \frac{0.2791953179185249767080279070796677842736 + \left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z}{\sqrt{3.350343815022303939343828460550867021084 + \left(6.012459259764103336465268512256443500519 + z\right) \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \end{array}\]
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 -4230301041346655513100977098981376:\\
\;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\

\mathbf{elif}\;z \le 203544506.3211925327777862548828125:\\
\;\;\;\;x + \frac{y}{\sqrt{3.350343815022303939343828460550867021084 + \left(6.012459259764103336465268512256443500519 + z\right) \cdot z}} \cdot \frac{0.2791953179185249767080279070796677842736 + \left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z}{\sqrt{3.350343815022303939343828460550867021084 + \left(6.012459259764103336465268512256443500519 + z\right) \cdot z}}\\

\mathbf{else}:\\
\;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\

\end{array}
double f(double x, double y, double z) {
        double r23141781 = x;
        double r23141782 = y;
        double r23141783 = z;
        double r23141784 = 0.0692910599291889;
        double r23141785 = r23141783 * r23141784;
        double r23141786 = 0.4917317610505968;
        double r23141787 = r23141785 + r23141786;
        double r23141788 = r23141787 * r23141783;
        double r23141789 = 0.279195317918525;
        double r23141790 = r23141788 + r23141789;
        double r23141791 = r23141782 * r23141790;
        double r23141792 = 6.012459259764103;
        double r23141793 = r23141783 + r23141792;
        double r23141794 = r23141793 * r23141783;
        double r23141795 = 3.350343815022304;
        double r23141796 = r23141794 + r23141795;
        double r23141797 = r23141791 / r23141796;
        double r23141798 = r23141781 + r23141797;
        return r23141798;
}
↓
double f(double x, double y, double z) {
        double r23141799 = z;
        double r23141800 = -4.2303010413466555e+33;
        bool r23141801 = r23141799 <= r23141800;
        double r23141802 = 0.0692910599291889;
        double r23141803 = y;
        double r23141804 = r23141802 * r23141803;
        double r23141805 = r23141803 / r23141799;
        double r23141806 = 0.07512208616047561;
        double r23141807 = r23141805 * r23141806;
        double r23141808 = r23141804 + r23141807;
        double r23141809 = x;
        double r23141810 = r23141808 + r23141809;
        double r23141811 = 203544506.32119253;
        bool r23141812 = r23141799 <= r23141811;
        double r23141813 = 3.350343815022304;
        double r23141814 = 6.012459259764103;
        double r23141815 = r23141814 + r23141799;
        double r23141816 = r23141815 * r23141799;
        double r23141817 = r23141813 + r23141816;
        double r23141818 = sqrt(r23141817);
        double r23141819 = r23141803 / r23141818;
        double r23141820 = 0.279195317918525;
        double r23141821 = 0.4917317610505968;
        double r23141822 = r23141799 * r23141802;
        double r23141823 = r23141821 + r23141822;
        double r23141824 = r23141823 * r23141799;
        double r23141825 = r23141820 + r23141824;
        double r23141826 = r23141825 / r23141818;
        double r23141827 = r23141819 * r23141826;
        double r23141828 = r23141809 + r23141827;
        double r23141829 = r23141812 ? r23141828 : r23141810;
        double r23141830 = r23141801 ? r23141810 : r23141829;
        return r23141830;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.1
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 < -4.2303010413466555e+33 or 203544506.32119253 < z

    1. Initial program 42.7

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

    2. Taylor expanded around inf 0.0

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

    if -4.2303010413466555e+33 < z < 203544506.32119253

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.7

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

    4. Applied times-frac0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4230301041346655513100977098981376:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \mathbf{elif}\;z \le 203544506.3211925327777862548828125:\\ \;\;\;\;x + \frac{y}{\sqrt{3.350343815022303939343828460550867021084 + \left(6.012459259764103336465268512256443500519 + z\right) \cdot z}} \cdot \frac{0.2791953179185249767080279070796677842736 + \left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z}{\sqrt{3.350343815022303939343828460550867021084 + \left(6.012459259764103336465268512256443500519 + z\right) \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))