Average Error: 19.8 → 0.1
Time: 1.0m
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -853081.36506114923395216464996337890625:\\ \;\;\;\;x + y \cdot \left(0.06929105992918889456166908757950295694172 + \left(\frac{0.07512208616047560960637952121032867580652}{z} - \frac{0.4046220386999212492717958866705885156989}{z \cdot z}\right)\right)\\ \mathbf{elif}\;z \le 665218.019642971456050872802734375:\\ \;\;\;\;x + \frac{z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736}{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.06929105992918889456166908757950295694172 + \left(\frac{0.07512208616047560960637952121032867580652}{z} - \frac{0.4046220386999212492717958866705885156989}{z \cdot z}\right)\right)\\ \end{array}\]
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 -853081.36506114923395216464996337890625:\\
\;\;\;\;x + y \cdot \left(0.06929105992918889456166908757950295694172 + \left(\frac{0.07512208616047560960637952121032867580652}{z} - \frac{0.4046220386999212492717958866705885156989}{z \cdot z}\right)\right)\\

\mathbf{elif}\;z \le 665218.019642971456050872802734375:\\
\;\;\;\;x + \frac{z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736}{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084} \cdot y\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \left(0.06929105992918889456166908757950295694172 + \left(\frac{0.07512208616047560960637952121032867580652}{z} - \frac{0.4046220386999212492717958866705885156989}{z \cdot z}\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r17430043 = x;
        double r17430044 = y;
        double r17430045 = z;
        double r17430046 = 0.0692910599291889;
        double r17430047 = r17430045 * r17430046;
        double r17430048 = 0.4917317610505968;
        double r17430049 = r17430047 + r17430048;
        double r17430050 = r17430049 * r17430045;
        double r17430051 = 0.279195317918525;
        double r17430052 = r17430050 + r17430051;
        double r17430053 = r17430044 * r17430052;
        double r17430054 = 6.012459259764103;
        double r17430055 = r17430045 + r17430054;
        double r17430056 = r17430055 * r17430045;
        double r17430057 = 3.350343815022304;
        double r17430058 = r17430056 + r17430057;
        double r17430059 = r17430053 / r17430058;
        double r17430060 = r17430043 + r17430059;
        return r17430060;
}
↓
double f(double x, double y, double z) {
        double r17430061 = z;
        double r17430062 = -853081.3650611492;
        bool r17430063 = r17430061 <= r17430062;
        double r17430064 = x;
        double r17430065 = y;
        double r17430066 = 0.0692910599291889;
        double r17430067 = 0.07512208616047561;
        double r17430068 = r17430067 / r17430061;
        double r17430069 = 0.40462203869992125;
        double r17430070 = r17430061 * r17430061;
        double r17430071 = r17430069 / r17430070;
        double r17430072 = r17430068 - r17430071;
        double r17430073 = r17430066 + r17430072;
        double r17430074 = r17430065 * r17430073;
        double r17430075 = r17430064 + r17430074;
        double r17430076 = 665218.0196429715;
        bool r17430077 = r17430061 <= r17430076;
        double r17430078 = r17430066 * r17430061;
        double r17430079 = 0.4917317610505968;
        double r17430080 = r17430078 + r17430079;
        double r17430081 = r17430061 * r17430080;
        double r17430082 = 0.279195317918525;
        double r17430083 = r17430081 + r17430082;
        double r17430084 = 6.012459259764103;
        double r17430085 = r17430084 + r17430061;
        double r17430086 = r17430061 * r17430085;
        double r17430087 = 3.350343815022304;
        double r17430088 = r17430086 + r17430087;
        double r17430089 = r17430083 / r17430088;
        double r17430090 = r17430089 * r17430065;
        double r17430091 = r17430064 + r17430090;
        double r17430092 = r17430077 ? r17430091 : r17430075;
        double r17430093 = r17430063 ? r17430075 : r17430092;
        return r17430093;
}

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

Original19.8
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 < -853081.3650611492 or 665218.0196429715 < z

    1. Initial program 40.3

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

    3. Applied *-un-lft-identity40.3

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

    4. Applied times-frac32.4

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

    5. Simplified32.4

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

    6. Taylor expanded around inf 0.0

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.06929105992918889456166908757950295694172 + 0.07512208616047560960637952121032867580652 \cdot \frac{1}{z}\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{1}{{z}^{2}}\right)}\]

    7. Simplified0.0

      \[\leadsto x + y \cdot \color{blue}{\left(0.06929105992918889456166908757950295694172 + \left(\frac{0.07512208616047560960637952121032867580652}{z} - \frac{0.4046220386999212492717958866705885156989}{z \cdot z}\right)\right)}\]

    if -853081.3650611492 < z < 665218.0196429715

    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 *-un-lft-identity0.1

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -853081.36506114923395216464996337890625:\\ \;\;\;\;x + y \cdot \left(0.06929105992918889456166908757950295694172 + \left(\frac{0.07512208616047560960637952121032867580652}{z} - \frac{0.4046220386999212492717958866705885156989}{z \cdot z}\right)\right)\\ \mathbf{elif}\;z \le 665218.019642971456050872802734375:\\ \;\;\;\;x + \frac{z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736}{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.06929105992918889456166908757950295694172 + \left(\frac{0.07512208616047560960637952121032867580652}{z} - \frac{0.4046220386999212492717958866705885156989}{z \cdot z}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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))))