Average Error: 20.1 → 0.1
Time: 23.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -305854124920.404541015625 \lor \neg \left(z \le 1558.167654028546394329168833792209625244\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\\ \end{array}\]
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 -305854124920.404541015625 \lor \neg \left(z \le 1558.167654028546394329168833792209625244\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r231593 = x;
        double r231594 = y;
        double r231595 = z;
        double r231596 = 0.0692910599291889;
        double r231597 = r231595 * r231596;
        double r231598 = 0.4917317610505968;
        double r231599 = r231597 + r231598;
        double r231600 = r231599 * r231595;
        double r231601 = 0.279195317918525;
        double r231602 = r231600 + r231601;
        double r231603 = r231594 * r231602;
        double r231604 = 6.012459259764103;
        double r231605 = r231595 + r231604;
        double r231606 = r231605 * r231595;
        double r231607 = 3.350343815022304;
        double r231608 = r231606 + r231607;
        double r231609 = r231603 / r231608;
        double r231610 = r231593 + r231609;
        return r231610;
}
↓
double f(double x, double y, double z) {
        double r231611 = z;
        double r231612 = -305854124920.40454;
        bool r231613 = r231611 <= r231612;
        double r231614 = 1558.1676540285464;
        bool r231615 = r231611 <= r231614;
        double r231616 = !r231615;
        bool r231617 = r231613 || r231616;
        double r231618 = y;
        double r231619 = r231618 / r231611;
        double r231620 = 0.07512208616047561;
        double r231621 = 0.0692910599291889;
        double r231622 = x;
        double r231623 = fma(r231618, r231621, r231622);
        double r231624 = fma(r231619, r231620, r231623);
        double r231625 = 6.012459259764103;
        double r231626 = r231611 + r231625;
        double r231627 = 3.350343815022304;
        double r231628 = fma(r231626, r231611, r231627);
        double r231629 = r231618 / r231628;
        double r231630 = 0.4917317610505968;
        double r231631 = fma(r231611, r231621, r231630);
        double r231632 = 0.279195317918525;
        double r231633 = fma(r231631, r231611, r231632);
        double r231634 = fma(r231629, r231633, r231622);
        double r231635 = r231617 ? r231624 : r231634;
        return r231635;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.1
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 < -305854124920.40454 or 1558.1676540285464 < z

    1. Initial program 41.3

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

    2. Simplified34.1

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.0

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

    if -305854124920.40454 < z < 1558.1676540285464

    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. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -305854124920.404541015625 \lor \neg \left(z \le 1558.167654028546394329168833792209625244\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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 -8120153.6524566747) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (/ 1 (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394))))