Average Error: 6.1 → 4.2
Time: 9.4s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 7689132349616876:\\ \;\;\;\;\sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]
\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}\;x \le 7689132349616876:\\
\;\;\;\;\sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x, double y, double z) {
        double r424815 = x;
        double r424816 = 0.5;
        double r424817 = r424815 - r424816;
        double r424818 = log(r424815);
        double r424819 = r424817 * r424818;
        double r424820 = r424819 - r424815;
        double r424821 = 0.91893853320467;
        double r424822 = r424820 + r424821;
        double r424823 = y;
        double r424824 = 0.0007936500793651;
        double r424825 = r424823 + r424824;
        double r424826 = z;
        double r424827 = r424825 * r424826;
        double r424828 = 0.0027777777777778;
        double r424829 = r424827 - r424828;
        double r424830 = r424829 * r424826;
        double r424831 = 0.083333333333333;
        double r424832 = r424830 + r424831;
        double r424833 = r424832 / r424815;
        double r424834 = r424822 + r424833;
        return r424834;
}
↓
double f(double x, double y, double z) {
        double r424835 = x;
        double r424836 = 7689132349616876.0;
        bool r424837 = r424835 <= r424836;
        double r424838 = 0.5;
        double r424839 = r424835 - r424838;
        double r424840 = log(r424835);
        double r424841 = r424839 * r424840;
        double r424842 = r424841 - r424835;
        double r424843 = 0.91893853320467;
        double r424844 = r424842 + r424843;
        double r424845 = sqrt(r424844);
        double r424846 = r424845 * r424845;
        double r424847 = y;
        double r424848 = 0.0007936500793651;
        double r424849 = r424847 + r424848;
        double r424850 = z;
        double r424851 = r424849 * r424850;
        double r424852 = 0.0027777777777778;
        double r424853 = r424851 - r424852;
        double r424854 = r424853 * r424850;
        double r424855 = 0.083333333333333;
        double r424856 = r424854 + r424855;
        double r424857 = r424856 / r424835;
        double r424858 = r424846 + r424857;
        double r424859 = 2.0;
        double r424860 = pow(r424850, r424859);
        double r424861 = r424860 / r424835;
        double r424862 = r424848 * r424861;
        double r424863 = 1.0;
        double r424864 = r424863 / r424835;
        double r424865 = log(r424864);
        double r424866 = fma(r424865, r424835, r424835);
        double r424867 = r424862 - r424866;
        double r424868 = fma(r424861, r424847, r424867);
        double r424869 = r424837 ? r424858 : r424868;
        return r424869;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.1
Target1.2
Herbie4.2
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 7689132349616876.0

    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. Using strategy rm

    3. Applied add-sqr-sqrt0.3

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

    if 7689132349616876.0 < x

    1. Initial program 10.6

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

    2. Simplified10.5

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

    3. Taylor expanded around inf 10.7

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

    4. Simplified7.2

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

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 7689132349616876:\\ \;\;\;\;\sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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)))