Average Error: 20.0 → 0.4
Time: 4.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -173066918.0464952 \lor \neg \left(z \le 1.075597970485127 \cdot 10^{-7}\right):\\ \;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \sqrt{\frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\right) \cdot \sqrt{\frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\\ \end{array}\]
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 -173066918.0464952 \lor \neg \left(z \le 1.075597970485127 \cdot 10^{-7}\right):\\
\;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r364749 = x;
        double r364750 = y;
        double r364751 = z;
        double r364752 = 0.0692910599291889;
        double r364753 = r364751 * r364752;
        double r364754 = 0.4917317610505968;
        double r364755 = r364753 + r364754;
        double r364756 = r364755 * r364751;
        double r364757 = 0.279195317918525;
        double r364758 = r364756 + r364757;
        double r364759 = r364750 * r364758;
        double r364760 = 6.012459259764103;
        double r364761 = r364751 + r364760;
        double r364762 = r364761 * r364751;
        double r364763 = 3.350343815022304;
        double r364764 = r364762 + r364763;
        double r364765 = r364759 / r364764;
        double r364766 = r364749 + r364765;
        return r364766;
}
↓
double f(double x, double y, double z) {
        double r364767 = z;
        double r364768 = -173066918.0464952;
        bool r364769 = r364767 <= r364768;
        double r364770 = 1.075597970485127e-07;
        bool r364771 = r364767 <= r364770;
        double r364772 = !r364771;
        bool r364773 = r364769 || r364772;
        double r364774 = x;
        double r364775 = 0.07512208616047561;
        double r364776 = y;
        double r364777 = r364776 / r364767;
        double r364778 = r364775 * r364777;
        double r364779 = 0.0692910599291889;
        double r364780 = r364779 * r364776;
        double r364781 = r364778 + r364780;
        double r364782 = r364774 + r364781;
        double r364783 = r364767 * r364779;
        double r364784 = 0.4917317610505968;
        double r364785 = r364783 + r364784;
        double r364786 = r364785 * r364767;
        double r364787 = 0.279195317918525;
        double r364788 = r364786 + r364787;
        double r364789 = 6.012459259764103;
        double r364790 = r364767 + r364789;
        double r364791 = r364790 * r364767;
        double r364792 = 3.350343815022304;
        double r364793 = r364791 + r364792;
        double r364794 = r364788 / r364793;
        double r364795 = sqrt(r364794);
        double r364796 = r364776 * r364795;
        double r364797 = r364796 * r364795;
        double r364798 = r364774 + r364797;
        double r364799 = r364773 ? r364782 : r364798;
        return r364799;
}

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

Original20.0
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -173066918.0464952 or 1.075597970485127e-07 < z

    1. Initial program 39.9

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

    2. Taylor expanded around inf 0.6

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

    if -173066918.0464952 < z < 1.075597970485127e-07

    1. Initial program 0.2

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

    3. Applied *-un-lft-identity0.2

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\color{blue}{1 \cdot \left(\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394\right)}}\]

    4. Applied times-frac0.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\]

    5. Simplified0.1

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

    7. Applied add-sqr-sqrt0.2

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

    8. Applied associate-*r*0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -173066918.0464952 \lor \neg \left(z \le 1.075597970485127 \cdot 10^{-7}\right):\\ \;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \sqrt{\frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\right) \cdot \sqrt{\frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))