Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C

Average Error: 26.6 → 1.0

Time: 8.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.054541883482835 \cdot 10^{64} \lor \neg \left(x \le 2485685511.02154016\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r508854 = x;
        double r508855 = 2.0;
        double r508856 = r508854 - r508855;
        double r508857 = 4.16438922228;
        double r508858 = r508854 * r508857;
        double r508859 = 78.6994924154;
        double r508860 = r508858 + r508859;
        double r508861 = r508860 * r508854;
        double r508862 = 137.519416416;
        double r508863 = r508861 + r508862;
        double r508864 = r508863 * r508854;
        double r508865 = y;
        double r508866 = r508864 + r508865;
        double r508867 = r508866 * r508854;
        double r508868 = z;
        double r508869 = r508867 + r508868;
        double r508870 = r508856 * r508869;
        double r508871 = 43.3400022514;
        double r508872 = r508854 + r508871;
        double r508873 = r508872 * r508854;
        double r508874 = 263.505074721;
        double r508875 = r508873 + r508874;
        double r508876 = r508875 * r508854;
        double r508877 = 313.399215894;
        double r508878 = r508876 + r508877;
        double r508879 = r508878 * r508854;
        double r508880 = 47.066876606;
        double r508881 = r508879 + r508880;
        double r508882 = r508870 / r508881;
        return r508882;
}

↓

double f(double x, double y, double z) {
        double r508883 = x;
        double r508884 = -1.054541883482835e+64;
        bool r508885 = r508883 <= r508884;
        double r508886 = 2485685511.02154;
        bool r508887 = r508883 <= r508886;
        double r508888 = !r508887;
        bool r508889 = r508885 || r508888;
        double r508890 = 4.16438922228;
        double r508891 = y;
        double r508892 = 2.0;
        double r508893 = pow(r508883, r508892);
        double r508894 = r508891 / r508893;
        double r508895 = 110.1139242984811;
        double r508896 = r508894 - r508895;
        double r508897 = fma(r508883, r508890, r508896);
        double r508898 = 2.0;
        double r508899 = r508883 - r508898;
        double r508900 = 1.0;
        double r508901 = 78.6994924154;
        double r508902 = fma(r508883, r508890, r508901);
        double r508903 = 137.519416416;
        double r508904 = fma(r508902, r508883, r508903);
        double r508905 = fma(r508904, r508883, r508891);
        double r508906 = z;
        double r508907 = fma(r508905, r508883, r508906);
        double r508908 = r508900 * r508907;
        double r508909 = 43.3400022514;
        double r508910 = r508883 + r508909;
        double r508911 = 263.505074721;
        double r508912 = fma(r508910, r508883, r508911);
        double r508913 = 313.399215894;
        double r508914 = fma(r508912, r508883, r508913);
        double r508915 = 47.066876606;
        double r508916 = fma(r508914, r508883, r508915);
        double r508917 = r508908 / r508916;
        double r508918 = r508899 * r508917;
        double r508919 = r508889 ? r508897 : r508918;
        return r508919;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.6
Target	0.5
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if x < -1.054541883482835e+64 or 2485685511.02154 < x

   1. Initial program 58.7

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

   2. Simplified 55.2

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

   3. Taylor expanded around inf 1.6

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

   4. Simplified 1.6

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

   if -1.054541883482835e+64 < x < 2485685511.02154

   1. Initial program 1.5

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

   2. Simplified 0.7

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

   4. Applied pow1 0.7

      \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\color{blue}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right)\right)}^{1}}, x, z\right)}}\]
   5. Using strategy rm

   6. Applied div-inv 0.7

      \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right)\right)}^{1}, x, z\right)}}}\]

   7. Simplified 0.5

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.054541883482835 \cdot 10^{64} \lor \neg \left(x \le 2485685511.02154016\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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)))