Average Error: 26.3 → 0.9
Time: 25.3s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.2489554391470123 \cdot 10^{30} \lor \neg \left(x \le 3.71248631582250911 \cdot 10^{28}\right):\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999964, x, \frac{y}{{x}^{2}}\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922227999964, x \cdot x, \mathsf{fma}\left(78.6994924154000017, x, 137.51941641600001\right)\right), x, y\right), x, z\right)}}\\ \end{array}\]
\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 -2.2489554391470123 \cdot 10^{30} \lor \neg \left(x \le 3.71248631582250911 \cdot 10^{28}\right):\\
\;\;\;\;\mathsf{fma}\left(4.16438922227999964, x, \frac{y}{{x}^{2}}\right) - 110.11392429848109\\

\mathbf{else}:\\
\;\;\;\;\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(4.16438922227999964, x \cdot x, \mathsf{fma}\left(78.6994924154000017, x, 137.51941641600001\right)\right), x, y\right), x, z\right)}}\\

\end{array}
double f(double x, double y, double z) {
        double r291730 = x;
        double r291731 = 2.0;
        double r291732 = r291730 - r291731;
        double r291733 = 4.16438922228;
        double r291734 = r291730 * r291733;
        double r291735 = 78.6994924154;
        double r291736 = r291734 + r291735;
        double r291737 = r291736 * r291730;
        double r291738 = 137.519416416;
        double r291739 = r291737 + r291738;
        double r291740 = r291739 * r291730;
        double r291741 = y;
        double r291742 = r291740 + r291741;
        double r291743 = r291742 * r291730;
        double r291744 = z;
        double r291745 = r291743 + r291744;
        double r291746 = r291732 * r291745;
        double r291747 = 43.3400022514;
        double r291748 = r291730 + r291747;
        double r291749 = r291748 * r291730;
        double r291750 = 263.505074721;
        double r291751 = r291749 + r291750;
        double r291752 = r291751 * r291730;
        double r291753 = 313.399215894;
        double r291754 = r291752 + r291753;
        double r291755 = r291754 * r291730;
        double r291756 = 47.066876606;
        double r291757 = r291755 + r291756;
        double r291758 = r291746 / r291757;
        return r291758;
}
↓
double f(double x, double y, double z) {
        double r291759 = x;
        double r291760 = -2.2489554391470123e+30;
        bool r291761 = r291759 <= r291760;
        double r291762 = 3.712486315822509e+28;
        bool r291763 = r291759 <= r291762;
        double r291764 = !r291763;
        bool r291765 = r291761 || r291764;
        double r291766 = 4.16438922228;
        double r291767 = y;
        double r291768 = 2.0;
        double r291769 = pow(r291759, r291768);
        double r291770 = r291767 / r291769;
        double r291771 = fma(r291766, r291759, r291770);
        double r291772 = 110.1139242984811;
        double r291773 = r291771 - r291772;
        double r291774 = 2.0;
        double r291775 = r291759 - r291774;
        double r291776 = 43.3400022514;
        double r291777 = r291759 + r291776;
        double r291778 = 263.505074721;
        double r291779 = fma(r291777, r291759, r291778);
        double r291780 = 313.399215894;
        double r291781 = fma(r291779, r291759, r291780);
        double r291782 = 47.066876606;
        double r291783 = fma(r291781, r291759, r291782);
        double r291784 = r291759 * r291759;
        double r291785 = 78.6994924154;
        double r291786 = 137.519416416;
        double r291787 = fma(r291785, r291759, r291786);
        double r291788 = fma(r291766, r291784, r291787);
        double r291789 = fma(r291788, r291759, r291767);
        double r291790 = z;
        double r291791 = fma(r291789, r291759, r291790);
        double r291792 = r291783 / r291791;
        double r291793 = r291775 / r291792;
        double r291794 = r291765 ? r291773 : r291793;
        return r291794;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.3
Target0.5
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.2489554391470123e+30 or 3.712486315822509e+28 < x

    1. Initial program 58.1

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

    2. Simplified54.0

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

    3. Taylor expanded around inf 1.5

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

    4. Simplified1.4

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

    if -2.2489554391470123e+30 < x < 3.712486315822509e+28

    1. Initial program 0.5

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

    2. Simplified0.5

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

    3. Taylor expanded around 0 0.5

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

    4. Simplified0.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.2489554391470123 \cdot 10^{30} \lor \neg \left(x \le 3.71248631582250911 \cdot 10^{28}\right):\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999964, x, \frac{y}{{x}^{2}}\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922227999964, x \cdot x, \mathsf{fma}\left(78.6994924154000017, x, 137.51941641600001\right)\right), x, y\right), x, z\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1.0) (/ (+ (* (+ (* (+ (* (+ (* 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 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))