Average Error: 19.3 → 0.1
Time: 7.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2712764.504135835 \lor \neg \left(z \le 63485.5636438174624\right):\\ \;\;\;\;x + y \cdot \left(\left(0.0692910599291888946 - \frac{0.404622038699921249}{z \cdot z}\right) + \frac{0.07512208616047561}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right) + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 -2712764.504135835 \lor \neg \left(z \le 63485.5636438174624\right):\\
\;\;\;\;x + y \cdot \left(\left(0.0692910599291888946 - \frac{0.404622038699921249}{z \cdot z}\right) + \frac{0.07512208616047561}{z}\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 r131582 = x;
        double r131583 = y;
        double r131584 = z;
        double r131585 = 0.0692910599291889;
        double r131586 = r131584 * r131585;
        double r131587 = 0.4917317610505968;
        double r131588 = r131586 + r131587;
        double r131589 = r131588 * r131584;
        double r131590 = 0.279195317918525;
        double r131591 = r131589 + r131590;
        double r131592 = r131583 * r131591;
        double r131593 = 6.012459259764103;
        double r131594 = r131584 + r131593;
        double r131595 = r131594 * r131584;
        double r131596 = 3.350343815022304;
        double r131597 = r131595 + r131596;
        double r131598 = r131592 / r131597;
        double r131599 = r131582 + r131598;
        return r131599;
}
↓
double f(double x, double y, double z) {
        double r131600 = z;
        double r131601 = -2712764.504135835;
        bool r131602 = r131600 <= r131601;
        double r131603 = 63485.56364381746;
        bool r131604 = r131600 <= r131603;
        double r131605 = !r131604;
        bool r131606 = r131602 || r131605;
        double r131607 = x;
        double r131608 = y;
        double r131609 = 0.0692910599291889;
        double r131610 = 0.40462203869992125;
        double r131611 = r131600 * r131600;
        double r131612 = r131610 / r131611;
        double r131613 = r131609 - r131612;
        double r131614 = 0.07512208616047561;
        double r131615 = r131614 / r131600;
        double r131616 = r131613 + r131615;
        double r131617 = r131608 * r131616;
        double r131618 = r131607 + r131617;
        double r131619 = r131600 * r131609;
        double r131620 = 0.4917317610505968;
        double r131621 = r131619 + r131620;
        double r131622 = cbrt(r131621);
        double r131623 = r131622 * r131622;
        double r131624 = r131622 * r131600;
        double r131625 = r131623 * r131624;
        double r131626 = 0.279195317918525;
        double r131627 = r131625 + r131626;
        double r131628 = 6.012459259764103;
        double r131629 = r131600 + r131628;
        double r131630 = r131629 * r131600;
        double r131631 = 3.350343815022304;
        double r131632 = r131630 + r131631;
        double r131633 = r131627 / r131632;
        double r131634 = r131608 * r131633;
        double r131635 = r131607 + r131634;
        double r131636 = r131606 ? r131618 : r131635;
        return r131636;
}

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

Original19.3
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 < -2712764.504135835 or 63485.56364381746 < z

    1. Initial program 39.5

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

    3. Applied *-un-lft-identity39.5

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

    4. Applied times-frac31.5

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

    5. Simplified31.5

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

    6. Taylor expanded around inf 0.0

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.07512208616047561 \cdot \frac{1}{z} + 0.0692910599291888946\right) - 0.404622038699921249 \cdot \frac{1}{{z}^{2}}\right)}\]

    7. Simplified0.0

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

    if -2712764.504135835 < z < 63485.56364381746

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2712764.504135835 \lor \neg \left(z \le 63485.5636438174624\right):\\ \;\;\;\;x + y \cdot \left(\left(0.0692910599291888946 - \frac{0.404622038699921249}{z \cdot z}\right) + \frac{0.07512208616047561}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot \sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679}\right) \cdot \left(\sqrt[3]{z \cdot 0.0692910599291888946 + 0.49173176105059679} \cdot z\right) + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))