Average Error: 20.4 → 0.2
Time: 1.0m
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2222553190536753720752001253376 \lor \neg \left(z \le 646280.956370810978114604949951171875\right):\\ \;\;\;\;x + \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(0.4917317610505967939715787906607147306204 \cdot z + 0.06929105992918889456166908757950295694172 \cdot {z}^{2}\right) + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]
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 -2222553190536753720752001253376 \lor \neg \left(z \le 646280.956370810978114604949951171875\right):\\
\;\;\;\;x + \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y \cdot \left(\left(0.4917317610505967939715787906607147306204 \cdot z + 0.06929105992918889456166908757950295694172 \cdot {z}^{2}\right) + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\

\end{array}
double f(double x, double y, double z) {
        double r288545 = x;
        double r288546 = y;
        double r288547 = z;
        double r288548 = 0.0692910599291889;
        double r288549 = r288547 * r288548;
        double r288550 = 0.4917317610505968;
        double r288551 = r288549 + r288550;
        double r288552 = r288551 * r288547;
        double r288553 = 0.279195317918525;
        double r288554 = r288552 + r288553;
        double r288555 = r288546 * r288554;
        double r288556 = 6.012459259764103;
        double r288557 = r288547 + r288556;
        double r288558 = r288557 * r288547;
        double r288559 = 3.350343815022304;
        double r288560 = r288558 + r288559;
        double r288561 = r288555 / r288560;
        double r288562 = r288545 + r288561;
        return r288562;
}
↓
double f(double x, double y, double z) {
        double r288563 = z;
        double r288564 = -2.2225531905367537e+30;
        bool r288565 = r288563 <= r288564;
        double r288566 = 646280.956370811;
        bool r288567 = r288563 <= r288566;
        double r288568 = !r288567;
        bool r288569 = r288565 || r288568;
        double r288570 = x;
        double r288571 = 0.07512208616047561;
        double r288572 = y;
        double r288573 = r288572 / r288563;
        double r288574 = r288571 * r288573;
        double r288575 = 0.0692910599291889;
        double r288576 = r288575 * r288572;
        double r288577 = r288574 + r288576;
        double r288578 = 0.40462203869992125;
        double r288579 = 2.0;
        double r288580 = pow(r288563, r288579);
        double r288581 = r288572 / r288580;
        double r288582 = r288578 * r288581;
        double r288583 = r288577 - r288582;
        double r288584 = r288570 + r288583;
        double r288585 = 0.4917317610505968;
        double r288586 = r288585 * r288563;
        double r288587 = r288575 * r288580;
        double r288588 = r288586 + r288587;
        double r288589 = 0.279195317918525;
        double r288590 = r288588 + r288589;
        double r288591 = r288572 * r288590;
        double r288592 = 6.012459259764103;
        double r288593 = r288563 + r288592;
        double r288594 = r288593 * r288563;
        double r288595 = 3.350343815022304;
        double r288596 = r288594 + r288595;
        double r288597 = r288591 / r288596;
        double r288598 = r288570 + r288597;
        double r288599 = r288569 ? r288584 : r288598;
        return r288599;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.4
Target0.1
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 < -2.2225531905367537e+30 or 646280.956370811 < z

    1. Initial program 42.2

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

    2. Taylor expanded around inf 0.0

      \[\leadsto x + \color{blue}{\left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)}\]

    if -2.2225531905367537e+30 < z < 646280.956370811

    1. Initial program 0.3

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

    2. Taylor expanded around 0 0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2222553190536753720752001253376 \lor \neg \left(z \le 646280.956370810978114604949951171875\right):\\ \;\;\;\;x + \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(0.4917317610505967939715787906607147306204 \cdot z + 0.06929105992918889456166908757950295694172 \cdot {z}^{2}\right) + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(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))))