Average Error: 20.1 → 0.1
Time: 4.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3370149036.10117674 \lor \neg \left(z \le 175959582.81735757\right):\\ \;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right) + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 -3370149036.10117674 \lor \neg \left(z \le 175959582.81735757\right):\\
\;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right) + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\

\end{array}
double f(double x, double y, double z) {
        double r326618 = x;
        double r326619 = y;
        double r326620 = z;
        double r326621 = 0.0692910599291889;
        double r326622 = r326620 * r326621;
        double r326623 = 0.4917317610505968;
        double r326624 = r326622 + r326623;
        double r326625 = r326624 * r326620;
        double r326626 = 0.279195317918525;
        double r326627 = r326625 + r326626;
        double r326628 = r326619 * r326627;
        double r326629 = 6.012459259764103;
        double r326630 = r326620 + r326629;
        double r326631 = r326630 * r326620;
        double r326632 = 3.350343815022304;
        double r326633 = r326631 + r326632;
        double r326634 = r326628 / r326633;
        double r326635 = r326618 + r326634;
        return r326635;
}
↓
double f(double x, double y, double z) {
        double r326636 = z;
        double r326637 = -3370149036.1011767;
        bool r326638 = r326636 <= r326637;
        double r326639 = 175959582.81735757;
        bool r326640 = r326636 <= r326639;
        double r326641 = !r326640;
        bool r326642 = r326638 || r326641;
        double r326643 = x;
        double r326644 = 0.07512208616047561;
        double r326645 = y;
        double r326646 = r326645 / r326636;
        double r326647 = r326644 * r326646;
        double r326648 = 0.0692910599291889;
        double r326649 = r326648 * r326645;
        double r326650 = r326647 + r326649;
        double r326651 = r326643 + r326650;
        double r326652 = r326636 * r326648;
        double r326653 = 0.4917317610505968;
        double r326654 = r326652 + r326653;
        double r326655 = cbrt(r326654);
        double r326656 = r326655 * r326655;
        double r326657 = r326655 * r326636;
        double r326658 = r326656 * r326657;
        double r326659 = 0.279195317918525;
        double r326660 = r326658 + r326659;
        double r326661 = 6.012459259764103;
        double r326662 = r326636 + r326661;
        double r326663 = r326662 * r326636;
        double r326664 = 3.350343815022304;
        double r326665 = r326663 + r326664;
        double r326666 = r326660 / r326665;
        double r326667 = r326645 * r326666;
        double r326668 = r326643 + r326667;
        double r326669 = r326642 ? r326651 : r326668;
        return r326669;
}

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

Derivation

  1. Split input into 2 regimes

  2. if z < -3370149036.1011767 or 175959582.81735757 < z

    1. Initial program 41.3

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

    2. Taylor expanded around inf 0.0

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

    if -3370149036.1011767 < z < 175959582.81735757

    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-cube-cbrt0.1

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

    8. Applied associate-*l*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3370149036.10117674 \lor \neg \left(z \le 175959582.81735757\right):\\ \;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right) + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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))))