Average Error: 19.6 → 0.1
Time: 11.2s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -255355953485.46405:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, x\right)\right)\\ \mathbf{elif}\;z \le 291922397.4765716:\\ \;\;\;\;x + \frac{0.279195317918525 + \left(0.4917317610505968 + 0.0692910599291889 \cdot z\right) \cdot z}{\sqrt{\left(6.012459259764103 + z\right) \cdot z + 3.350343815022304}} \cdot \frac{y}{\sqrt{\left(6.012459259764103 + z\right) \cdot z + 3.350343815022304}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, x\right)\right)\\ \end{array}\]
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 -255355953485.46405:\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, x\right)\right)\\

\mathbf{elif}\;z \le 291922397.4765716:\\
\;\;\;\;x + \frac{0.279195317918525 + \left(0.4917317610505968 + 0.0692910599291889 \cdot z\right) \cdot z}{\sqrt{\left(6.012459259764103 + z\right) \cdot z + 3.350343815022304}} \cdot \frac{y}{\sqrt{\left(6.012459259764103 + z\right) \cdot z + 3.350343815022304}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, x\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r6461496 = x;
        double r6461497 = y;
        double r6461498 = z;
        double r6461499 = 0.0692910599291889;
        double r6461500 = r6461498 * r6461499;
        double r6461501 = 0.4917317610505968;
        double r6461502 = r6461500 + r6461501;
        double r6461503 = r6461502 * r6461498;
        double r6461504 = 0.279195317918525;
        double r6461505 = r6461503 + r6461504;
        double r6461506 = r6461497 * r6461505;
        double r6461507 = 6.012459259764103;
        double r6461508 = r6461498 + r6461507;
        double r6461509 = r6461508 * r6461498;
        double r6461510 = 3.350343815022304;
        double r6461511 = r6461509 + r6461510;
        double r6461512 = r6461506 / r6461511;
        double r6461513 = r6461496 + r6461512;
        return r6461513;
}
↓
double f(double x, double y, double z) {
        double r6461514 = z;
        double r6461515 = -255355953485.46405;
        bool r6461516 = r6461514 <= r6461515;
        double r6461517 = 0.0692910599291889;
        double r6461518 = y;
        double r6461519 = r6461518 / r6461514;
        double r6461520 = 0.07512208616047561;
        double r6461521 = x;
        double r6461522 = fma(r6461519, r6461520, r6461521);
        double r6461523 = fma(r6461517, r6461518, r6461522);
        double r6461524 = 291922397.4765716;
        bool r6461525 = r6461514 <= r6461524;
        double r6461526 = 0.279195317918525;
        double r6461527 = 0.4917317610505968;
        double r6461528 = r6461517 * r6461514;
        double r6461529 = r6461527 + r6461528;
        double r6461530 = r6461529 * r6461514;
        double r6461531 = r6461526 + r6461530;
        double r6461532 = 6.012459259764103;
        double r6461533 = r6461532 + r6461514;
        double r6461534 = r6461533 * r6461514;
        double r6461535 = 3.350343815022304;
        double r6461536 = r6461534 + r6461535;
        double r6461537 = sqrt(r6461536);
        double r6461538 = r6461531 / r6461537;
        double r6461539 = r6461518 / r6461537;
        double r6461540 = r6461538 * r6461539;
        double r6461541 = r6461521 + r6461540;
        double r6461542 = r6461525 ? r6461541 : r6461523;
        double r6461543 = r6461516 ? r6461523 : r6461542;
        return r6461543;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.6
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 < -255355953485.46405 or 291922397.4765716 < z

    1. Initial program 40.4

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

    2. Simplified33.8

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

    4. Applied clear-num33.9

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

    5. Taylor expanded around inf 0.0

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

    6. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, x\right)\right)}\]

    if -255355953485.46405 < z < 291922397.4765716

    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 add-sqr-sqrt0.5

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

    4. Applied times-frac0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -255355953485.46405:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, x\right)\right)\\ \mathbf{elif}\;z \le 291922397.4765716:\\ \;\;\;\;x + \frac{0.279195317918525 + \left(0.4917317610505968 + 0.0692910599291889 \cdot z\right) \cdot z}{\sqrt{\left(6.012459259764103 + z\right) \cdot z + 3.350343815022304}} \cdot \frac{y}{\sqrt{\left(6.012459259764103 + z\right) \cdot z + 3.350343815022304}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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))))