Average Error: 20.2 → 0.1
Time: 18.4s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.110187933824355434057475957993495901711 \cdot 10^{45} \lor \neg \left(z \le 267536142.104578495025634765625\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}\\ \end{array}\]
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 -5.110187933824355434057475957993495901711 \cdot 10^{45} \lor \neg \left(z \le 267536142.104578495025634765625\right):\\
\;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \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)}\\

\end{array}
double f(double x, double y, double z) {
        double r219805 = x;
        double r219806 = y;
        double r219807 = z;
        double r219808 = 0.0692910599291889;
        double r219809 = r219807 * r219808;
        double r219810 = 0.4917317610505968;
        double r219811 = r219809 + r219810;
        double r219812 = r219811 * r219807;
        double r219813 = 0.279195317918525;
        double r219814 = r219812 + r219813;
        double r219815 = r219806 * r219814;
        double r219816 = 6.012459259764103;
        double r219817 = r219807 + r219816;
        double r219818 = r219817 * r219807;
        double r219819 = 3.350343815022304;
        double r219820 = r219818 + r219819;
        double r219821 = r219815 / r219820;
        double r219822 = r219805 + r219821;
        return r219822;
}
↓
double f(double x, double y, double z) {
        double r219823 = z;
        double r219824 = -5.1101879338243554e+45;
        bool r219825 = r219823 <= r219824;
        double r219826 = 267536142.1045785;
        bool r219827 = r219823 <= r219826;
        double r219828 = !r219827;
        bool r219829 = r219825 || r219828;
        double r219830 = 0.07512208616047561;
        double r219831 = y;
        double r219832 = r219831 / r219823;
        double r219833 = 0.0692910599291889;
        double r219834 = x;
        double r219835 = fma(r219831, r219833, r219834);
        double r219836 = fma(r219830, r219832, r219835);
        double r219837 = 0.4917317610505968;
        double r219838 = fma(r219823, r219833, r219837);
        double r219839 = 0.279195317918525;
        double r219840 = fma(r219838, r219823, r219839);
        double r219841 = 6.012459259764103;
        double r219842 = r219823 + r219841;
        double r219843 = 3.350343815022304;
        double r219844 = fma(r219842, r219823, r219843);
        double r219845 = r219840 / r219844;
        double r219846 = r219831 * r219845;
        double r219847 = r219834 + r219846;
        double r219848 = r219829 ? r219836 : r219847;
        return r219848;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.2
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 < -5.1101879338243554e+45 or 267536142.1045785 < z

    1. Initial program 43.7

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

    2. Simplified36.6

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -5.1101879338243554e+45 < z < 267536142.1045785

    1. Initial program 0.4

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

    3. Applied *-un-lft-identity0.4

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

    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

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.110187933824355434057475957993495901711 \cdot 10^{45} \lor \neg \left(z \le 267536142.104578495025634765625\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1 (+ (* (+ 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))))