Average Error: 5.7 → 3.9
Time: 8.7s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le -4.55339649112024007 \cdot 10^{26} \lor \neg \left(\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 12325115026455952\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{\left(\sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)} \cdot z\right) + 0.0833333333333329956}{x} - \left(x - 0.91893853320467001\right)\right)\\ \end{array}\]
\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}
↓
\begin{array}{l}
\mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le -4.55339649112024007 \cdot 10^{26} \lor \neg \left(\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 12325115026455952\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{\left(\sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)} \cdot z\right) + 0.0833333333333329956}{x} - \left(x - 0.91893853320467001\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r1159906 = x;
        double r1159907 = 0.5;
        double r1159908 = r1159906 - r1159907;
        double r1159909 = log(r1159906);
        double r1159910 = r1159908 * r1159909;
        double r1159911 = r1159910 - r1159906;
        double r1159912 = 0.91893853320467;
        double r1159913 = r1159911 + r1159912;
        double r1159914 = y;
        double r1159915 = 0.0007936500793651;
        double r1159916 = r1159914 + r1159915;
        double r1159917 = z;
        double r1159918 = r1159916 * r1159917;
        double r1159919 = 0.0027777777777778;
        double r1159920 = r1159918 - r1159919;
        double r1159921 = r1159920 * r1159917;
        double r1159922 = 0.083333333333333;
        double r1159923 = r1159921 + r1159922;
        double r1159924 = r1159923 / r1159906;
        double r1159925 = r1159913 + r1159924;
        return r1159925;
}
↓
double f(double x, double y, double z) {
        double r1159926 = y;
        double r1159927 = 0.0007936500793651;
        double r1159928 = r1159926 + r1159927;
        double r1159929 = z;
        double r1159930 = r1159928 * r1159929;
        double r1159931 = 0.0027777777777778;
        double r1159932 = r1159930 - r1159931;
        double r1159933 = r1159932 * r1159929;
        double r1159934 = -4.55339649112024e+26;
        bool r1159935 = r1159933 <= r1159934;
        double r1159936 = 12325115026455952.0;
        bool r1159937 = r1159933 <= r1159936;
        double r1159938 = !r1159937;
        bool r1159939 = r1159935 || r1159938;
        double r1159940 = 2.0;
        double r1159941 = pow(r1159929, r1159940);
        double r1159942 = x;
        double r1159943 = r1159941 / r1159942;
        double r1159944 = r1159927 * r1159943;
        double r1159945 = 1.0;
        double r1159946 = r1159945 / r1159942;
        double r1159947 = log(r1159946);
        double r1159948 = fma(r1159947, r1159942, r1159942);
        double r1159949 = r1159944 - r1159948;
        double r1159950 = fma(r1159943, r1159926, r1159949);
        double r1159951 = log(r1159942);
        double r1159952 = 0.5;
        double r1159953 = r1159942 - r1159952;
        double r1159954 = -r1159931;
        double r1159955 = fma(r1159928, r1159929, r1159954);
        double r1159956 = cbrt(r1159955);
        double r1159957 = r1159956 * r1159956;
        double r1159958 = r1159956 * r1159929;
        double r1159959 = r1159957 * r1159958;
        double r1159960 = 0.083333333333333;
        double r1159961 = r1159959 + r1159960;
        double r1159962 = r1159961 / r1159942;
        double r1159963 = 0.91893853320467;
        double r1159964 = r1159942 - r1159963;
        double r1159965 = r1159962 - r1159964;
        double r1159966 = fma(r1159951, r1159953, r1159965);
        double r1159967 = r1159939 ? r1159950 : r1159966;
        return r1159967;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original5.7
Target1.1
Herbie3.9
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < -4.55339649112024e+26 or 12325115026455952.0 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)

    1. Initial program 16.9

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    2. Simplified16.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - 0.5, \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x} - \left(x - 0.91893853320467001\right)\right)}\]

    3. Taylor expanded around inf 17.2

      \[\leadsto \color{blue}{\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - \left(x + x \cdot \log \left(\frac{1}{x}\right)\right)}\]

    4. Simplified11.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)}\]

    if -4.55339649112024e+26 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < 12325115026455952.0

    1. Initial program 0.2

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    2. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - 0.5, \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x} - \left(x - 0.91893853320467001\right)\right)}\]
    3. Using strategy rm

    4. Applied fma-neg0.2

      \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \frac{\color{blue}{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)} \cdot z + 0.0833333333333329956}{x} - \left(x - 0.91893853320467001\right)\right)\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.2

      \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \frac{\color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)}\right)} \cdot z + 0.0833333333333329956}{x} - \left(x - 0.91893853320467001\right)\right)\]

    7. Applied associate-*l*0.2

      \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)} \cdot z\right)} + 0.0833333333333329956}{x} - \left(x - 0.91893853320467001\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le -4.55339649112024007 \cdot 10^{26} \lor \neg \left(\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 12325115026455952\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{\left(\sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y + 7.93650079365100015 \cdot 10^{-4}, z, -0.0027777777777778\right)} \cdot z\right) + 0.0833333333333329956}{x} - \left(x - 0.91893853320467001\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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)))