Average Error: 6.1 → 4.6
Time: 25.4s
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 r350929 = x;
        double r350930 = 0.5;
        double r350931 = r350929 - r350930;
        double r350932 = log(r350929);
        double r350933 = r350931 * r350932;
        double r350934 = r350933 - r350929;
        double r350935 = 0.91893853320467;
        double r350936 = r350934 + r350935;
        double r350937 = y;
        double r350938 = 0.0007936500793651;
        double r350939 = r350937 + r350938;
        double r350940 = z;
        double r350941 = r350939 * r350940;
        double r350942 = 0.0027777777777778;
        double r350943 = r350941 - r350942;
        double r350944 = r350943 * r350940;
        double r350945 = 0.083333333333333;
        double r350946 = r350944 + r350945;
        double r350947 = r350946 / r350929;
        double r350948 = r350936 + r350947;
        return r350948;
}
↓
double f(double x, double y, double z) {
        double r350949 = x;
        double r350950 = 1.4885262816105155e+82;
        bool r350951 = r350949 <= r350950;
        double r350952 = 0.91893853320467;
        double r350953 = log(r350949);
        double r350954 = 0.5;
        double r350955 = r350949 - r350954;
        double r350956 = 1.0;
        double r350957 = y;
        double r350958 = 0.0007936500793651;
        double r350959 = r350957 + r350958;
        double r350960 = z;
        double r350961 = r350959 * r350960;
        double r350962 = 0.0027777777777778;
        double r350963 = r350961 - r350962;
        double r350964 = 0.083333333333333;
        double r350965 = fma(r350963, r350960, r350964);
        double r350966 = r350949 / r350965;
        double r350967 = r350956 / r350966;
        double r350968 = fma(r350953, r350955, r350967);
        double r350969 = r350952 + r350968;
        double r350970 = r350969 - r350949;
        double r350971 = r350960 / r350949;
        double r350972 = -r350971;
        double r350973 = 2.0;
        double r350974 = pow(r350960, r350973);
        double r350975 = r350974 / r350949;
        double r350976 = r350975 * r350959;
        double r350977 = fma(r350962, r350972, r350976);
        double r350978 = fma(r350953, r350955, r350977);
        double r350979 = r350952 + r350978;
        double r350980 = r350979 - r350949;
        double r350981 = r350951 ? r350970 : r350980;
        return r350981;
}

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)))