Average Error: 6.1 → 4.1
Time: 27.8s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.58175379650287135512985768710700713416 \cdot 10^{45}:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \mathsf{fma}\left(x - 0.5, \log \left(\sqrt{x}\right), 0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]
\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 2.58175379650287135512985768710700713416 \cdot 10^{45}:\\
\;\;\;\;\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \mathsf{fma}\left(x - 0.5, \log \left(\sqrt{x}\right), 0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r456760 = x;
        double r456761 = 0.5;
        double r456762 = r456760 - r456761;
        double r456763 = log(r456760);
        double r456764 = r456762 * r456763;
        double r456765 = r456764 - r456760;
        double r456766 = 0.91893853320467;
        double r456767 = r456765 + r456766;
        double r456768 = y;
        double r456769 = 0.0007936500793651;
        double r456770 = r456768 + r456769;
        double r456771 = z;
        double r456772 = r456770 * r456771;
        double r456773 = 0.0027777777777778;
        double r456774 = r456772 - r456773;
        double r456775 = r456774 * r456771;
        double r456776 = 0.083333333333333;
        double r456777 = r456775 + r456776;
        double r456778 = r456777 / r456760;
        double r456779 = r456767 + r456778;
        return r456779;
}
↓
double f(double x, double y, double z) {
        double r456780 = x;
        double r456781 = 2.5817537965028714e+45;
        bool r456782 = r456780 <= r456781;
        double r456783 = 0.5;
        double r456784 = r456780 - r456783;
        double r456785 = sqrt(r456780);
        double r456786 = log(r456785);
        double r456787 = r456784 * r456786;
        double r456788 = 0.91893853320467;
        double r456789 = r456788 - r456780;
        double r456790 = fma(r456784, r456786, r456789);
        double r456791 = r456787 + r456790;
        double r456792 = y;
        double r456793 = 0.0007936500793651;
        double r456794 = r456792 + r456793;
        double r456795 = z;
        double r456796 = r456794 * r456795;
        double r456797 = 0.0027777777777778;
        double r456798 = r456796 - r456797;
        double r456799 = r456798 * r456795;
        double r456800 = 0.083333333333333;
        double r456801 = r456799 + r456800;
        double r456802 = r456801 / r456780;
        double r456803 = r456791 + r456802;
        double r456804 = log(r456780);
        double r456805 = r456784 * r456804;
        double r456806 = r456805 - r456780;
        double r456807 = r456806 + r456788;
        double r456808 = 2.0;
        double r456809 = pow(r456795, r456808);
        double r456810 = r456809 / r456780;
        double r456811 = r456810 * r456794;
        double r456812 = r456795 / r456780;
        double r456813 = r456797 * r456812;
        double r456814 = r456811 - r456813;
        double r456815 = r456807 + r456814;
        double r456816 = r456782 ? r456803 : r456815;
        return r456816;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < 2.5817537965028714e+45

    1. Initial program 0.4

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

    3. Applied add-sqr-sqrt0.4

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

    4. Applied log-prod0.4

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

    5. Applied distribute-lft-in0.4

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

    6. Applied associate--l+0.4

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

    7. Applied associate-+l+0.4

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

    8. Simplified0.4

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

    if 2.5817537965028714e+45 < x

    1. Initial program 11.4

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

    2. Taylor expanded around inf 11.5

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

    3. Simplified7.6

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

  4. Final simplification4.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.58175379650287135512985768710700713416 \cdot 10^{45}:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \mathsf{fma}\left(x - 0.5, \log \left(\sqrt{x}\right), 0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))