Average Error: 19.5 → 0.1
Time: 56.6s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2257324487.703424:\\ \;\;\;\;x + y \cdot \left(\left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right) - \frac{0.40462203869992125}{z \cdot z}\right)\\ \mathbf{elif}\;z \le 23415.342663009633:\\ \;\;\;\;\frac{\frac{z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(6.012459259764103 + z\right) + 3.350343815022304}}}{\sqrt{z \cdot \left(6.012459259764103 + z\right) + 3.350343815022304}} \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right) - \frac{0.40462203869992125}{z \cdot z}\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2257324487.703424:\\
\;\;\;\;x + y \cdot \left(\left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right) - \frac{0.40462203869992125}{z \cdot z}\right)\\

\mathbf{elif}\;z \le 23415.342663009633:\\
\;\;\;\;\frac{\frac{z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(6.012459259764103 + z\right) + 3.350343815022304}}}{\sqrt{z \cdot \left(6.012459259764103 + z\right) + 3.350343815022304}} \cdot y + x\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \left(\left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right) - \frac{0.40462203869992125}{z \cdot z}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r17907461 = x;
        double r17907462 = y;
        double r17907463 = z;
        double r17907464 = 0.0692910599291889;
        double r17907465 = r17907463 * r17907464;
        double r17907466 = 0.4917317610505968;
        double r17907467 = r17907465 + r17907466;
        double r17907468 = r17907467 * r17907463;
        double r17907469 = 0.279195317918525;
        double r17907470 = r17907468 + r17907469;
        double r17907471 = r17907462 * r17907470;
        double r17907472 = 6.012459259764103;
        double r17907473 = r17907463 + r17907472;
        double r17907474 = r17907473 * r17907463;
        double r17907475 = 3.350343815022304;
        double r17907476 = r17907474 + r17907475;
        double r17907477 = r17907471 / r17907476;
        double r17907478 = r17907461 + r17907477;
        return r17907478;
}
↓
double f(double x, double y, double z) {
        double r17907479 = z;
        double r17907480 = -2257324487.703424;
        bool r17907481 = r17907479 <= r17907480;
        double r17907482 = x;
        double r17907483 = y;
        double r17907484 = 0.0692910599291889;
        double r17907485 = 0.07512208616047561;
        double r17907486 = r17907485 / r17907479;
        double r17907487 = r17907484 + r17907486;
        double r17907488 = 0.40462203869992125;
        double r17907489 = r17907479 * r17907479;
        double r17907490 = r17907488 / r17907489;
        double r17907491 = r17907487 - r17907490;
        double r17907492 = r17907483 * r17907491;
        double r17907493 = r17907482 + r17907492;
        double r17907494 = 23415.342663009633;
        bool r17907495 = r17907479 <= r17907494;
        double r17907496 = r17907484 * r17907479;
        double r17907497 = 0.4917317610505968;
        double r17907498 = r17907496 + r17907497;
        double r17907499 = r17907479 * r17907498;
        double r17907500 = 0.279195317918525;
        double r17907501 = r17907499 + r17907500;
        double r17907502 = 6.012459259764103;
        double r17907503 = r17907502 + r17907479;
        double r17907504 = r17907479 * r17907503;
        double r17907505 = 3.350343815022304;
        double r17907506 = r17907504 + r17907505;
        double r17907507 = sqrt(r17907506);
        double r17907508 = r17907501 / r17907507;
        double r17907509 = r17907508 / r17907507;
        double r17907510 = r17907509 * r17907483;
        double r17907511 = r17907510 + r17907482;
        double r17907512 = r17907495 ? r17907511 : r17907493;
        double r17907513 = r17907481 ? r17907493 : r17907512;
        return r17907513;
}

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

Original19.5
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.652456675:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2257324487.703424 or 23415.342663009633 < z

    1. Initial program 40.0

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity40.0

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\color{blue}{1 \cdot \left(\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304\right)}}\]

    4. Applied times-frac32.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\]

    5. Simplified32.1

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

    6. Taylor expanded around inf 0.0

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right) - 0.40462203869992125 \cdot \frac{1}{{z}^{2}}\right)}\]

    7. Simplified0.0

      \[\leadsto x + y \cdot \color{blue}{\left(\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) - \frac{0.40462203869992125}{z \cdot z}\right)}\]

    if -2257324487.703424 < z < 23415.342663009633

    1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
    2. Using strategy rm

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

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\color{blue}{1 \cdot \left(\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304\right)}}\]

    4. Applied times-frac0.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\]

    5. Simplified0.1

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt0.5

      \[\leadsto x + y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\color{blue}{\sqrt{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \cdot \sqrt{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}}\]

    8. Applied associate-/r*0.2

      \[\leadsto x + y \cdot \color{blue}{\frac{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\sqrt{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}}{\sqrt{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2257324487.703424:\\ \;\;\;\;x + y \cdot \left(\left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right) - \frac{0.40462203869992125}{z \cdot z}\right)\\ \mathbf{elif}\;z \le 23415.342663009633:\\ \;\;\;\;\frac{\frac{z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(6.012459259764103 + z\right) + 3.350343815022304}}}{\sqrt{z \cdot \left(6.012459259764103 + z\right) + 3.350343815022304}} \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right) - \frac{0.40462203869992125}{z \cdot z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ 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))))