Average Error: 20.2 → 0.2
Time: 4.1s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.7686121240438417 \cdot 10^{45} \lor \neg \left(z \le 0.487999161623279887\right):\\ \;\;\;\;x + y \cdot \left(\left(0.07512208616047561 \cdot \frac{1}{z} + 0.0692910599291888946\right) - 0.404622038699921249 \cdot \frac{1}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 -8.7686121240438417 \cdot 10^{45} \lor \neg \left(z \le 0.487999161623279887\right):\\
\;\;\;\;x + y \cdot \left(\left(0.07512208616047561 \cdot \frac{1}{z} + 0.0692910599291888946\right) - 0.404622038699921249 \cdot \frac{1}{{z}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \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 r391477 = x;
        double r391478 = y;
        double r391479 = z;
        double r391480 = 0.0692910599291889;
        double r391481 = r391479 * r391480;
        double r391482 = 0.4917317610505968;
        double r391483 = r391481 + r391482;
        double r391484 = r391483 * r391479;
        double r391485 = 0.279195317918525;
        double r391486 = r391484 + r391485;
        double r391487 = r391478 * r391486;
        double r391488 = 6.012459259764103;
        double r391489 = r391479 + r391488;
        double r391490 = r391489 * r391479;
        double r391491 = 3.350343815022304;
        double r391492 = r391490 + r391491;
        double r391493 = r391487 / r391492;
        double r391494 = r391477 + r391493;
        return r391494;
}
↓
double f(double x, double y, double z) {
        double r391495 = z;
        double r391496 = -8.768612124043842e+45;
        bool r391497 = r391495 <= r391496;
        double r391498 = 0.4879991616232799;
        bool r391499 = r391495 <= r391498;
        double r391500 = !r391499;
        bool r391501 = r391497 || r391500;
        double r391502 = x;
        double r391503 = y;
        double r391504 = 0.07512208616047561;
        double r391505 = 1.0;
        double r391506 = r391505 / r391495;
        double r391507 = r391504 * r391506;
        double r391508 = 0.0692910599291889;
        double r391509 = r391507 + r391508;
        double r391510 = 0.40462203869992125;
        double r391511 = 2.0;
        double r391512 = pow(r391495, r391511);
        double r391513 = r391505 / r391512;
        double r391514 = r391510 * r391513;
        double r391515 = r391509 - r391514;
        double r391516 = r391503 * r391515;
        double r391517 = r391502 + r391516;
        double r391518 = r391495 * r391508;
        double r391519 = 0.4917317610505968;
        double r391520 = r391518 + r391519;
        double r391521 = r391520 * r391495;
        double r391522 = 0.279195317918525;
        double r391523 = r391521 + r391522;
        double r391524 = 6.012459259764103;
        double r391525 = r391495 + r391524;
        double r391526 = r391525 * r391495;
        double r391527 = 3.350343815022304;
        double r391528 = r391526 + r391527;
        double r391529 = r391523 / r391528;
        double r391530 = r391503 * r391529;
        double r391531 = r391502 + r391530;
        double r391532 = r391501 ? r391517 : r391531;
        return r391532;
}

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

Original20.2
Target0.2
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 < -8.768612124043842e+45 or 0.4879991616232799 < z

    1. Initial program 42.9

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

    3. Applied *-un-lft-identity42.9

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

    4. Applied times-frac34.8

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

    5. Simplified34.8

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

    6. Taylor expanded around inf 0.2

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

    if -8.768612124043842e+45 < z < 0.4879991616232799

    1. Initial program 0.5

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

    3. Applied *-un-lft-identity0.5

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

    4. Applied times-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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.7686121240438417 \cdot 10^{45} \lor \neg \left(z \le 0.487999161623279887\right):\\ \;\;\;\;x + y \cdot \left(\left(0.07512208616047561 \cdot \frac{1}{z} + 0.0692910599291888946\right) - 0.404622038699921249 \cdot \frac{1}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))))