Average Error: 20.6 → 0.1
Time: 17.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -464704203.059111535549163818359375 \lor \neg \left(z \le 46621187.154576636850833892822265625\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736\right)}{3.350343815022303939343828460550867021084 + z \cdot \left(z + 6.012459259764103336465268512256443500519\right)}\\ \end{array}\]
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 \lor \neg \left(z \le 46621187.154576636850833892822265625\right):\\
\;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r219734 = x;
        double r219735 = y;
        double r219736 = z;
        double r219737 = 0.0692910599291889;
        double r219738 = r219736 * r219737;
        double r219739 = 0.4917317610505968;
        double r219740 = r219738 + r219739;
        double r219741 = r219740 * r219736;
        double r219742 = 0.279195317918525;
        double r219743 = r219741 + r219742;
        double r219744 = r219735 * r219743;
        double r219745 = 6.012459259764103;
        double r219746 = r219736 + r219745;
        double r219747 = r219746 * r219736;
        double r219748 = 3.350343815022304;
        double r219749 = r219747 + r219748;
        double r219750 = r219744 / r219749;
        double r219751 = r219734 + r219750;
        return r219751;
}
↓
double f(double x, double y, double z) {
        double r219752 = z;
        double r219753 = -464704203.05911154;
        bool r219754 = r219752 <= r219753;
        double r219755 = 46621187.15457664;
        bool r219756 = r219752 <= r219755;
        double r219757 = !r219756;
        bool r219758 = r219754 || r219757;
        double r219759 = 0.07512208616047561;
        double r219760 = y;
        double r219761 = r219760 / r219752;
        double r219762 = 0.0692910599291889;
        double r219763 = x;
        double r219764 = fma(r219762, r219760, r219763);
        double r219765 = fma(r219759, r219761, r219764);
        double r219766 = r219752 * r219762;
        double r219767 = 0.4917317610505968;
        double r219768 = r219766 + r219767;
        double r219769 = r219752 * r219768;
        double r219770 = 0.279195317918525;
        double r219771 = r219769 + r219770;
        double r219772 = r219760 * r219771;
        double r219773 = 3.350343815022304;
        double r219774 = 6.012459259764103;
        double r219775 = r219752 + r219774;
        double r219776 = r219752 * r219775;
        double r219777 = r219773 + r219776;
        double r219778 = r219772 / r219777;
        double r219779 = r219763 + r219778;
        double r219780 = r219758 ? r219765 : r219779;
        return r219780;
}

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. Simplified33.0

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

    4. Applied add-exp-log34.4

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

    5. Applied add-exp-log34.6

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

    6. Applied div-exp34.6

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

    7. Simplified33.0

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

    8. Taylor expanded around inf 0.0

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

    9. Simplified0.0

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

    if -464704203.05911154 < z < 46621187.15457664

    1. Initial program 0.1

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

  4. Final simplification0.1

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

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))))