Average Error: 26.5 → 0.8
Time: 8.6s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.5558868413567101 \cdot 10^{22} \lor \neg \left(x \le 1.1453649274389147 \cdot 10^{36}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.5558868413567101 \cdot 10^{22} \lor \neg \left(x \le 1.1453649274389147 \cdot 10^{36}\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\

\end{array}
double f(double x, double y, double z) {
        double r340586 = x;
        double r340587 = 2.0;
        double r340588 = r340586 - r340587;
        double r340589 = 4.16438922228;
        double r340590 = r340586 * r340589;
        double r340591 = 78.6994924154;
        double r340592 = r340590 + r340591;
        double r340593 = r340592 * r340586;
        double r340594 = 137.519416416;
        double r340595 = r340593 + r340594;
        double r340596 = r340595 * r340586;
        double r340597 = y;
        double r340598 = r340596 + r340597;
        double r340599 = r340598 * r340586;
        double r340600 = z;
        double r340601 = r340599 + r340600;
        double r340602 = r340588 * r340601;
        double r340603 = 43.3400022514;
        double r340604 = r340586 + r340603;
        double r340605 = r340604 * r340586;
        double r340606 = 263.505074721;
        double r340607 = r340605 + r340606;
        double r340608 = r340607 * r340586;
        double r340609 = 313.399215894;
        double r340610 = r340608 + r340609;
        double r340611 = r340610 * r340586;
        double r340612 = 47.066876606;
        double r340613 = r340611 + r340612;
        double r340614 = r340602 / r340613;
        return r340614;
}
↓
double f(double x, double y, double z) {
        double r340615 = x;
        double r340616 = -3.55588684135671e+22;
        bool r340617 = r340615 <= r340616;
        double r340618 = 1.1453649274389147e+36;
        bool r340619 = r340615 <= r340618;
        double r340620 = !r340619;
        bool r340621 = r340617 || r340620;
        double r340622 = 4.16438922228;
        double r340623 = y;
        double r340624 = 2.0;
        double r340625 = pow(r340615, r340624);
        double r340626 = r340623 / r340625;
        double r340627 = 110.1139242984811;
        double r340628 = r340626 - r340627;
        double r340629 = fma(r340615, r340622, r340628);
        double r340630 = 2.0;
        double r340631 = r340615 - r340630;
        double r340632 = r340615 * r340622;
        double r340633 = 78.6994924154;
        double r340634 = r340632 + r340633;
        double r340635 = r340634 * r340615;
        double r340636 = 137.519416416;
        double r340637 = r340635 + r340636;
        double r340638 = r340637 * r340615;
        double r340639 = r340638 + r340623;
        double r340640 = r340639 * r340615;
        double r340641 = z;
        double r340642 = r340640 + r340641;
        double r340643 = r340631 * r340642;
        double r340644 = 43.3400022514;
        double r340645 = r340615 + r340644;
        double r340646 = r340645 * r340615;
        double r340647 = 263.505074721;
        double r340648 = r340646 + r340647;
        double r340649 = r340648 * r340615;
        double r340650 = 313.399215894;
        double r340651 = r340649 + r340650;
        double r340652 = r340651 * r340615;
        double r340653 = 47.066876606;
        double r340654 = r340652 + r340653;
        double r340655 = r340643 / r340654;
        double r340656 = r340621 ? r340629 : r340655;
        return r340656;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.5
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.55588684135671e+22 or 1.1453649274389147e+36 < x

    1. Initial program 58.3

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

    2. Simplified53.7

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}}}\]

    3. Taylor expanded around inf 1.1

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]

    4. Simplified1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)}\]

    if -3.55588684135671e+22 < x < 1.1453649274389147e+36

    1. Initial program 0.6

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.5558868413567101 \cdot 10^{22} \lor \neg \left(x \le 1.1453649274389147 \cdot 10^{36}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))