Average Error: 19.5 → 0.1
Time: 9.4s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2052833865386989.75 \lor \neg \left(z \le 352885967.354919910430908203125\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]
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 -2052833865386989.75 \lor \neg \left(z \le 352885967.354919910430908203125\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;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}\\

\end{array}
double f(double x, double y, double z) {
        double r442631 = x;
        double r442632 = y;
        double r442633 = z;
        double r442634 = 0.0692910599291889;
        double r442635 = r442633 * r442634;
        double r442636 = 0.4917317610505968;
        double r442637 = r442635 + r442636;
        double r442638 = r442637 * r442633;
        double r442639 = 0.279195317918525;
        double r442640 = r442638 + r442639;
        double r442641 = r442632 * r442640;
        double r442642 = 6.012459259764103;
        double r442643 = r442633 + r442642;
        double r442644 = r442643 * r442633;
        double r442645 = 3.350343815022304;
        double r442646 = r442644 + r442645;
        double r442647 = r442641 / r442646;
        double r442648 = r442631 + r442647;
        return r442648;
}
↓
double f(double x, double y, double z) {
        double r442649 = z;
        double r442650 = -2052833865386989.8;
        bool r442651 = r442649 <= r442650;
        double r442652 = 352885967.3549199;
        bool r442653 = r442649 <= r442652;
        double r442654 = !r442653;
        bool r442655 = r442651 || r442654;
        double r442656 = y;
        double r442657 = r442656 / r442649;
        double r442658 = 0.07512208616047561;
        double r442659 = 0.0692910599291889;
        double r442660 = x;
        double r442661 = fma(r442659, r442656, r442660);
        double r442662 = fma(r442657, r442658, r442661);
        double r442663 = r442649 * r442659;
        double r442664 = 0.4917317610505968;
        double r442665 = r442663 + r442664;
        double r442666 = r442665 * r442649;
        double r442667 = 0.279195317918525;
        double r442668 = r442666 + r442667;
        double r442669 = r442656 * r442668;
        double r442670 = 6.012459259764103;
        double r442671 = r442649 + r442670;
        double r442672 = r442671 * r442649;
        double r442673 = 3.350343815022304;
        double r442674 = r442672 + r442673;
        double r442675 = r442669 / r442674;
        double r442676 = r442660 + r442675;
        double r442677 = r442655 ? r442662 : r442676;
        return r442677;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.5
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 < -2052833865386989.8 or 352885967.3549199 < z

    1. Initial program 41.4

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

    2. Simplified34.7

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

    4. Applied div-inv34.7

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

    5. Taylor expanded around inf 0.0

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

    6. Simplified0.0

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

    if -2052833865386989.8 < z < 352885967.3549199

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2052833865386989.75 \lor \neg \left(z \le 352885967.354919910430908203125\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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))))