Average Error: 20.0 → 0.2
Time: 19.3s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -123702675580259303424:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{elif}\;z \le 0.02992000358306671353725292306080518756062:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \end{array}\]
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 -123702675580259303424:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\

\mathbf{elif}\;z \le 0.02992000358306671353725292306080518756062:\\
\;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r14594567 = x;
        double r14594568 = y;
        double r14594569 = z;
        double r14594570 = 0.0692910599291889;
        double r14594571 = r14594569 * r14594570;
        double r14594572 = 0.4917317610505968;
        double r14594573 = r14594571 + r14594572;
        double r14594574 = r14594573 * r14594569;
        double r14594575 = 0.279195317918525;
        double r14594576 = r14594574 + r14594575;
        double r14594577 = r14594568 * r14594576;
        double r14594578 = 6.012459259764103;
        double r14594579 = r14594569 + r14594578;
        double r14594580 = r14594579 * r14594569;
        double r14594581 = 3.350343815022304;
        double r14594582 = r14594580 + r14594581;
        double r14594583 = r14594577 / r14594582;
        double r14594584 = r14594567 + r14594583;
        return r14594584;
}
↓
double f(double x, double y, double z) {
        double r14594585 = z;
        double r14594586 = -1.237026755802593e+20;
        bool r14594587 = r14594585 <= r14594586;
        double r14594588 = y;
        double r14594589 = r14594588 / r14594585;
        double r14594590 = 0.07512208616047561;
        double r14594591 = 0.0692910599291889;
        double r14594592 = x;
        double r14594593 = fma(r14594588, r14594591, r14594592);
        double r14594594 = fma(r14594589, r14594590, r14594593);
        double r14594595 = 0.029920003583066714;
        bool r14594596 = r14594585 <= r14594595;
        double r14594597 = 0.4917317610505968;
        double r14594598 = fma(r14594591, r14594585, r14594597);
        double r14594599 = 0.279195317918525;
        double r14594600 = fma(r14594598, r14594585, r14594599);
        double r14594601 = 6.012459259764103;
        double r14594602 = r14594601 + r14594585;
        double r14594603 = 3.350343815022304;
        double r14594604 = fma(r14594602, r14594585, r14594603);
        double r14594605 = r14594600 / r14594604;
        double r14594606 = fma(r14594605, r14594588, r14594592);
        double r14594607 = r14594596 ? r14594606 : r14594594;
        double r14594608 = r14594587 ? r14594594 : r14594607;
        return r14594608;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.0
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.237026755802593e+20 or 0.029920003583066714 < 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. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    if -1.237026755802593e+20 < z < 0.029920003583066714

    1. Initial program 0.2

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

    2. Simplified0.1

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

    3. Taylor expanded around 0 0.1

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\color{blue}{6.012459259764103336465268512256443500519 \cdot z + \left({z}^{2} + 3.350343815022303939343828460550867021084\right)}}, y, x\right)\]

    4. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -123702675580259303424:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{elif}\;z \le 0.02992000358306671353725292306080518756062:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))))