Average Error: 20.1 → 0.5
Time: 5.0s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -276447135.675921142 \lor \neg \left(z \le 1.257025260484713 \cdot 10^{-12}\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(\sqrt[3]{z + 6.0124592597641033} \cdot \sqrt[3]{z + 6.0124592597641033}\right) \cdot \left(\sqrt[3]{z + 6.0124592597641033} \cdot z\right) + 3.35034381502230394}\\ \end{array}\]
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 -276447135.675921142 \lor \neg \left(z \le 1.257025260484713 \cdot 10^{-12}\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 \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(\sqrt[3]{z + 6.0124592597641033} \cdot \sqrt[3]{z + 6.0124592597641033}\right) \cdot \left(\sqrt[3]{z + 6.0124592597641033} \cdot z\right) + 3.35034381502230394}\\

\end{array}
double f(double x, double y, double z) {
        double r277734 = x;
        double r277735 = y;
        double r277736 = z;
        double r277737 = 0.0692910599291889;
        double r277738 = r277736 * r277737;
        double r277739 = 0.4917317610505968;
        double r277740 = r277738 + r277739;
        double r277741 = r277740 * r277736;
        double r277742 = 0.279195317918525;
        double r277743 = r277741 + r277742;
        double r277744 = r277735 * r277743;
        double r277745 = 6.012459259764103;
        double r277746 = r277736 + r277745;
        double r277747 = r277746 * r277736;
        double r277748 = 3.350343815022304;
        double r277749 = r277747 + r277748;
        double r277750 = r277744 / r277749;
        double r277751 = r277734 + r277750;
        return r277751;
}
↓
double f(double x, double y, double z) {
        double r277752 = z;
        double r277753 = -276447135.67592114;
        bool r277754 = r277752 <= r277753;
        double r277755 = 1.257025260484713e-12;
        bool r277756 = r277752 <= r277755;
        double r277757 = !r277756;
        bool r277758 = r277754 || r277757;
        double r277759 = x;
        double r277760 = 0.07512208616047561;
        double r277761 = y;
        double r277762 = r277761 / r277752;
        double r277763 = r277760 * r277762;
        double r277764 = 0.0692910599291889;
        double r277765 = r277764 * r277761;
        double r277766 = r277763 + r277765;
        double r277767 = 0.40462203869992125;
        double r277768 = 2.0;
        double r277769 = pow(r277752, r277768);
        double r277770 = r277761 / r277769;
        double r277771 = r277767 * r277770;
        double r277772 = r277766 - r277771;
        double r277773 = r277759 + r277772;
        double r277774 = r277752 * r277764;
        double r277775 = 0.4917317610505968;
        double r277776 = r277774 + r277775;
        double r277777 = r277776 * r277752;
        double r277778 = 0.279195317918525;
        double r277779 = r277777 + r277778;
        double r277780 = r277761 * r277779;
        double r277781 = 6.012459259764103;
        double r277782 = r277752 + r277781;
        double r277783 = cbrt(r277782);
        double r277784 = r277783 * r277783;
        double r277785 = r277783 * r277752;
        double r277786 = r277784 * r277785;
        double r277787 = 3.350343815022304;
        double r277788 = r277786 + r277787;
        double r277789 = r277780 / r277788;
        double r277790 = r277759 + r277789;
        double r277791 = r277758 ? r277773 : r277790;
        return r277791;
}

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

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

Derivation

  1. Split input into 2 regimes

  2. if z < -276447135.67592114 or 1.257025260484713e-12 < z

    1. Initial program 39.4

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

    2. Taylor expanded around inf 0.9

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

    if -276447135.67592114 < z < 1.257025260484713e-12

    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-cube-cbrt0.1

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\color{blue}{\left(\left(\sqrt[3]{z + 6.0124592597641033} \cdot \sqrt[3]{z + 6.0124592597641033}\right) \cdot \sqrt[3]{z + 6.0124592597641033}\right)} \cdot z + 3.35034381502230394}\]

    4. Applied associate-*l*0.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -276447135.675921142 \lor \neg \left(z \le 1.257025260484713 \cdot 10^{-12}\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(\sqrt[3]{z + 6.0124592597641033} \cdot \sqrt[3]{z + 6.0124592597641033}\right) \cdot \left(\sqrt[3]{z + 6.0124592597641033} \cdot z\right) + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))))