Average Error: 6.1 → 4.6
Time: 26.4s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.488526281610515453770934858449448718954 \cdot 10^{82}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\frac{x}{\mathsf{fma}\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321, z, 0.08333333333333299564049667651488562114537\right)}}\right)\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(0.002777777777777800001512975569539776188321, -\frac{z}{x}, \frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right)\right)\right)\right) - x\\ \end{array}\]
\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 1.488526281610515453770934858449448718954 \cdot 10^{82}:\\
\;\;\;\;\left(0.9189385332046700050057097541866824030876 + \mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\frac{x}{\mathsf{fma}\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321, z, 0.08333333333333299564049667651488562114537\right)}}\right)\right) - x\\

\mathbf{else}:\\
\;\;\;\;\left(0.9189385332046700050057097541866824030876 + \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(0.002777777777777800001512975569539776188321, -\frac{z}{x}, \frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right)\right)\right)\right) - x\\

\end{array}
double f(double x, double y, double z) {
        double r307831 = x;
        double r307832 = 0.5;
        double r307833 = r307831 - r307832;
        double r307834 = log(r307831);
        double r307835 = r307833 * r307834;
        double r307836 = r307835 - r307831;
        double r307837 = 0.91893853320467;
        double r307838 = r307836 + r307837;
        double r307839 = y;
        double r307840 = 0.0007936500793651;
        double r307841 = r307839 + r307840;
        double r307842 = z;
        double r307843 = r307841 * r307842;
        double r307844 = 0.0027777777777778;
        double r307845 = r307843 - r307844;
        double r307846 = r307845 * r307842;
        double r307847 = 0.083333333333333;
        double r307848 = r307846 + r307847;
        double r307849 = r307848 / r307831;
        double r307850 = r307838 + r307849;
        return r307850;
}
↓
double f(double x, double y, double z) {
        double r307851 = x;
        double r307852 = 1.4885262816105155e+82;
        bool r307853 = r307851 <= r307852;
        double r307854 = 0.91893853320467;
        double r307855 = log(r307851);
        double r307856 = 0.5;
        double r307857 = r307851 - r307856;
        double r307858 = 1.0;
        double r307859 = y;
        double r307860 = 0.0007936500793651;
        double r307861 = r307859 + r307860;
        double r307862 = z;
        double r307863 = r307861 * r307862;
        double r307864 = 0.0027777777777778;
        double r307865 = r307863 - r307864;
        double r307866 = 0.083333333333333;
        double r307867 = fma(r307865, r307862, r307866);
        double r307868 = r307851 / r307867;
        double r307869 = r307858 / r307868;
        double r307870 = fma(r307855, r307857, r307869);
        double r307871 = r307854 + r307870;
        double r307872 = r307871 - r307851;
        double r307873 = r307862 / r307851;
        double r307874 = -r307873;
        double r307875 = 2.0;
        double r307876 = pow(r307862, r307875);
        double r307877 = r307876 / r307851;
        double r307878 = r307877 * r307861;
        double r307879 = fma(r307864, r307874, r307878);
        double r307880 = fma(r307855, r307857, r307879);
        double r307881 = r307854 + r307880;
        double r307882 = r307881 - r307851;
        double r307883 = r307853 ? r307872 : r307882;
        return r307883;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.1
Target1.2
Herbie4.6
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{0.08333333333333299564049667651488562114537}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 1.4885262816105155e+82

    1. Initial program 0.8

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

    2. Simplified0.8

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

    4. Applied clear-num1.0

      \[\leadsto \left(0.9189385332046700050057097541866824030876 + \mathsf{fma}\left(\log x, x - 0.5, \color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321, z, 0.08333333333333299564049667651488562114537\right)}}}\right)\right) - x\]

    if 1.4885262816105155e+82 < x

    1. Initial program 12.6

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

    2. Simplified12.6

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

    3. Taylor expanded around inf 12.7

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

    4. Simplified9.1

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

  4. Final simplification4.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.488526281610515453770934858449448718954 \cdot 10^{82}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\frac{x}{\mathsf{fma}\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321, z, 0.08333333333333299564049667651488562114537\right)}}\right)\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(0.002777777777777800001512975569539776188321, -\frac{z}{x}, \frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right)\right)\right)\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :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)))