Average Error: 20.5 → 0.2
Time: 1.4m
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -128244192.27895916 \lor \neg \left(z \le 0.0137923017903093716\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}
↓
\begin{array}{l}
\mathbf{if}\;z \le -128244192.27895916 \lor \neg \left(z \le 0.0137923017903093716\right):\\
\;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\\

\end{array}
double f(double x, double y, double z) {
        double r297674 = x;
        double r297675 = y;
        double r297676 = z;
        double r297677 = 0.0692910599291889;
        double r297678 = r297676 * r297677;
        double r297679 = 0.4917317610505968;
        double r297680 = r297678 + r297679;
        double r297681 = r297680 * r297676;
        double r297682 = 0.279195317918525;
        double r297683 = r297681 + r297682;
        double r297684 = r297675 * r297683;
        double r297685 = 6.012459259764103;
        double r297686 = r297676 + r297685;
        double r297687 = r297686 * r297676;
        double r297688 = 3.350343815022304;
        double r297689 = r297687 + r297688;
        double r297690 = r297684 / r297689;
        double r297691 = r297674 + r297690;
        return r297691;
}
↓
double f(double x, double y, double z) {
        double r297692 = z;
        double r297693 = -128244192.27895916;
        bool r297694 = r297692 <= r297693;
        double r297695 = 0.013792301790309372;
        bool r297696 = r297692 <= r297695;
        double r297697 = !r297696;
        bool r297698 = r297694 || r297697;
        double r297699 = x;
        double r297700 = 0.07512208616047561;
        double r297701 = y;
        double r297702 = r297701 / r297692;
        double r297703 = r297700 * r297702;
        double r297704 = 0.0692910599291889;
        double r297705 = r297704 * r297701;
        double r297706 = r297703 + r297705;
        double r297707 = 0.40462203869992125;
        double r297708 = 2.0;
        double r297709 = pow(r297692, r297708);
        double r297710 = r297701 / r297709;
        double r297711 = r297707 * r297710;
        double r297712 = r297706 - r297711;
        double r297713 = r297699 + r297712;
        double r297714 = 6.012459259764103;
        double r297715 = r297692 + r297714;
        double r297716 = r297715 * r297692;
        double r297717 = 3.350343815022304;
        double r297718 = r297716 + r297717;
        double r297719 = sqrt(r297718);
        double r297720 = r297701 / r297719;
        double r297721 = r297692 * r297704;
        double r297722 = 0.4917317610505968;
        double r297723 = r297721 + r297722;
        double r297724 = r297723 * r297692;
        double r297725 = 0.279195317918525;
        double r297726 = r297724 + r297725;
        double r297727 = r297726 / r297719;
        double r297728 = r297720 * r297727;
        double r297729 = r297699 + r297728;
        double r297730 = r297698 ? r297713 : r297729;
        return r297730;
}

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

Original20.5
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -128244192.27895916 or 0.013792301790309372 < z

    1. Initial program 41.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

    2. Taylor expanded around inf 0.2

      \[\leadsto x + \color{blue}{\left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)}\]

    if -128244192.27895916 < z < 0.013792301790309372

    1. Initial program 0.1

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.5

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\color{blue}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394} \cdot \sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}}\]

    4. Applied times-frac0.2

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -128244192.27895916 \lor \neg \left(z \le 0.0137923017903093716\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\sqrt{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(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))))