Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Error: 5.9 → 0.3

Time: 19.7s

Precision: 64

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 29348759.7692681215703487396240234375:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log x - x\right) + \left(\frac{0.08333333333333299564049667651488562114537 + \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot \left(z \cdot z\right) - 0.002777777777777800001512975569539776188321 \cdot z\right)}{x} + 0.9189385332046700050057097541866824030876\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot \frac{z}{\frac{x}{z}} - \frac{0.002777777777777800001512975569539776188321 \cdot z}{x}\right)\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r429843 = x;
        double r429844 = 0.5;
        double r429845 = r429843 - r429844;
        double r429846 = log(r429843);
        double r429847 = r429845 * r429846;
        double r429848 = r429847 - r429843;
        double r429849 = 0.91893853320467;
        double r429850 = r429848 + r429849;
        double r429851 = y;
        double r429852 = 0.0007936500793651;
        double r429853 = r429851 + r429852;
        double r429854 = z;
        double r429855 = r429853 * r429854;
        double r429856 = 0.0027777777777778;
        double r429857 = r429855 - r429856;
        double r429858 = r429857 * r429854;
        double r429859 = 0.083333333333333;
        double r429860 = r429858 + r429859;
        double r429861 = r429860 / r429843;
        double r429862 = r429850 + r429861;
        return r429862;
}

↓

double f(double x, double y, double z) {
        double r429863 = x;
        double r429864 = 29348759.76926812;
        bool r429865 = r429863 <= r429864;
        double r429866 = 0.5;
        double r429867 = r429863 - r429866;
        double r429868 = log(r429863);
        double r429869 = r429867 * r429868;
        double r429870 = r429869 - r429863;
        double r429871 = 0.083333333333333;
        double r429872 = 0.0007936500793651;
        double r429873 = y;
        double r429874 = r429872 + r429873;
        double r429875 = z;
        double r429876 = r429875 * r429875;
        double r429877 = r429874 * r429876;
        double r429878 = 0.0027777777777778;
        double r429879 = r429878 * r429875;
        double r429880 = r429877 - r429879;
        double r429881 = r429871 + r429880;
        double r429882 = r429881 / r429863;
        double r429883 = 0.91893853320467;
        double r429884 = r429882 + r429883;
        double r429885 = r429870 + r429884;
        double r429886 = r429863 / r429875;
        double r429887 = r429875 / r429886;
        double r429888 = r429874 * r429887;
        double r429889 = r429879 / r429863;
        double r429890 = r429888 - r429889;
        double r429891 = r429883 + r429890;
        double r429892 = cbrt(r429863);
        double r429893 = log(r429892);
        double r429894 = r429893 * r429867;
        double r429895 = r429894 - r429863;
        double r429896 = r429892 * r429892;
        double r429897 = log(r429896);
        double r429898 = r429867 * r429897;
        double r429899 = r429895 + r429898;
        double r429900 = r429891 + r429899;
        double r429901 = r429865 ? r429885 : r429900;
        return r429901;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.2
Herbie	0.3

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{0.08333333333333299564049667651488562114537}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right)\]

Derivation

1. Split input into 2 regimes

2. if x < 29348759.76926812

   1. Initial program 0.1

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\left(\left(x - 0.5\right) \cdot \log x - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)}\]

   3. Taylor expanded around 0 0.1

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log x - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + \color{blue}{\left(\left({z}^{2} \cdot y + 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot {z}^{2}\right) - 0.002777777777777800001512975569539776188321 \cdot z\right)}}{x}\right)\]

   4. Simplified 0.1

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log x - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + \color{blue}{\left(\left(z \cdot z\right) \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - z \cdot 0.002777777777777800001512975569539776188321\right)}}{x}\right)\]

   if 29348759.76926812 < x

   1. Initial program 10.0

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   2. Simplified 10.0

      \[\leadsto \color{blue}{\left(\left(x - 0.5\right) \cdot \log x - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 10.0

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   5. Applied log-prod 10.1

      \[\leadsto \left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   6. Applied distribute-rgt-in 10.1

      \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right)} - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   7. Applied associate--l+ 10.0

      \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right)} + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   8. Taylor expanded around inf 10.2

      \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + \left(0.9189385332046700050057097541866824030876 + \color{blue}{\left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)}\right)\]

   9. Simplified 0.5

      \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + \left(0.9189385332046700050057097541866824030876 + \color{blue}{\left(\frac{z}{\frac{x}{z}} \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - \frac{z \cdot 0.002777777777777800001512975569539776188321}{x}\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 29348759.7692681215703487396240234375:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log x - x\right) + \left(\frac{0.08333333333333299564049667651488562114537 + \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot \left(z \cdot z\right) - 0.002777777777777800001512975569539776188321 \cdot z\right)}{x} + 0.9189385332046700050057097541866824030876\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot \frac{z}{\frac{x}{z}} - \frac{0.002777777777777800001512975569539776188321 \cdot z}{x}\right)\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))