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

Average Error: 27.2 → 0.5

Time: 13.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.0944108871381443 \cdot 10^{65} \lor \neg \left(x \le 7.2564691634128278 \cdot 10^{43}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.113924298481081\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r1834 = x;
        double r1835 = 2.0;
        double r1836 = r1834 - r1835;
        double r1837 = 4.16438922228;
        double r1838 = r1834 * r1837;
        double r1839 = 78.6994924154;
        double r1840 = r1838 + r1839;
        double r1841 = r1840 * r1834;
        double r1842 = 137.519416416;
        double r1843 = r1841 + r1842;
        double r1844 = r1843 * r1834;
        double r1845 = y;
        double r1846 = r1844 + r1845;
        double r1847 = r1846 * r1834;
        double r1848 = z;
        double r1849 = r1847 + r1848;
        double r1850 = r1836 * r1849;
        double r1851 = 43.3400022514;
        double r1852 = r1834 + r1851;
        double r1853 = r1852 * r1834;
        double r1854 = 263.505074721;
        double r1855 = r1853 + r1854;
        double r1856 = r1855 * r1834;
        double r1857 = 313.399215894;
        double r1858 = r1856 + r1857;
        double r1859 = r1858 * r1834;
        double r1860 = 47.066876606;
        double r1861 = r1859 + r1860;
        double r1862 = r1850 / r1861;
        return r1862;
}

↓

double f(double x, double y, double z) {
        double r1863 = x;
        double r1864 = -9.094410887138144e+65;
        bool r1865 = r1863 <= r1864;
        double r1866 = 7.256469163412828e+43;
        bool r1867 = r1863 <= r1866;
        double r1868 = !r1867;
        bool r1869 = r1865 || r1868;
        double r1870 = 4.16438922228;
        double r1871 = y;
        double r1872 = 2.0;
        double r1873 = pow(r1863, r1872);
        double r1874 = r1871 / r1873;
        double r1875 = 110.11392429848108;
        double r1876 = r1874 - r1875;
        double r1877 = fma(r1863, r1870, r1876);
        double r1878 = 2.0;
        double r1879 = r1863 - r1878;
        double r1880 = 1.0;
        double r1881 = 78.6994924154;
        double r1882 = fma(r1863, r1870, r1881);
        double r1883 = 137.519416416;
        double r1884 = fma(r1882, r1863, r1883);
        double r1885 = fma(r1884, r1863, r1871);
        double r1886 = z;
        double r1887 = fma(r1885, r1863, r1886);
        double r1888 = 43.3400022514;
        double r1889 = r1863 + r1888;
        double r1890 = 263.505074721;
        double r1891 = fma(r1889, r1863, r1890);
        double r1892 = 313.399215894;
        double r1893 = fma(r1891, r1863, r1892);
        double r1894 = 47.066876606;
        double r1895 = fma(r1893, r1863, r1894);
        double r1896 = r1887 / r1895;
        double r1897 = r1880 * r1896;
        double r1898 = r1879 * r1897;
        double r1899 = r1869 ? r1877 : r1898;
        return r1899;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.2
Target	0.4
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if x < -9.094410887138144e+65 or 7.256469163412828e+43 < x

   1. Initial program 62.4

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

   2. Simplified 59.2

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

   4. Applied flip-- 59.2

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

   5. Applied associate-/l/ 59.2

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

   6. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.113924298481081}\]

   7. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.113924298481081\right)}\]

   if -9.094410887138144e+65 < x < 7.256469163412828e+43

   1. Initial program 1.7

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

   2. Simplified 0.8

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

   4. Applied div-inv 0.8

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

   6. Applied *-un-lft-identity 0.8

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

   7. Applied *-un-lft-identity 0.8

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

   8. Applied times-frac 0.8

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

   9. Applied add-cube-cbrt 0.8

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

   10. Applied times-frac 0.8

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

   11. Simplified 0.8

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

   12. Simplified 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.0944108871381443 \cdot 10^{65} \lor \neg \left(x \le 7.2564691634128278 \cdot 10^{43}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.113924298481081\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))