Average Error: 5.9 → 4.0
Time: 8.1s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 7633786980962179:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]
\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\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 7633786980962179:\\
\;\;\;\;\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}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.936500793651000149400709382518925849581 \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 r501559 = x;
        double r501560 = 0.5;
        double r501561 = r501559 - r501560;
        double r501562 = log(r501559);
        double r501563 = r501561 * r501562;
        double r501564 = r501563 - r501559;
        double r501565 = 0.91893853320467;
        double r501566 = r501564 + r501565;
        double r501567 = y;
        double r501568 = 0.0007936500793651;
        double r501569 = r501567 + r501568;
        double r501570 = z;
        double r501571 = r501569 * r501570;
        double r501572 = 0.0027777777777778;
        double r501573 = r501571 - r501572;
        double r501574 = r501573 * r501570;
        double r501575 = 0.083333333333333;
        double r501576 = r501574 + r501575;
        double r501577 = r501576 / r501559;
        double r501578 = r501566 + r501577;
        return r501578;
}
↓
double f(double x, double y, double z) {
        double r501579 = x;
        double r501580 = 7633786980962179.0;
        bool r501581 = r501579 <= r501580;
        double r501582 = 0.5;
        double r501583 = r501579 - r501582;
        double r501584 = log(r501579);
        double r501585 = r501583 * r501584;
        double r501586 = r501585 - r501579;
        double r501587 = 0.91893853320467;
        double r501588 = r501586 + r501587;
        double r501589 = y;
        double r501590 = 0.0007936500793651;
        double r501591 = r501589 + r501590;
        double r501592 = z;
        double r501593 = r501591 * r501592;
        double r501594 = 0.0027777777777778;
        double r501595 = r501593 - r501594;
        double r501596 = r501595 * r501592;
        double r501597 = 0.083333333333333;
        double r501598 = r501596 + r501597;
        double r501599 = r501598 / r501579;
        double r501600 = r501588 + r501599;
        double r501601 = 2.0;
        double r501602 = pow(r501592, r501601);
        double r501603 = r501602 / r501579;
        double r501604 = r501590 * r501603;
        double r501605 = 1.0;
        double r501606 = r501605 / r501579;
        double r501607 = log(r501606);
        double r501608 = fma(r501607, r501579, r501579);
        double r501609 = r501604 - r501608;
        double r501610 = fma(r501603, r501589, r501609);
        double r501611 = r501581 ? r501600 : r501610;
        return r501611;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original5.9
Target1.2
Herbie4.0
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{0.08333333333333299564049667651488562114537}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 7633786980962179.0

    1. Initial program 0.2

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

    if 7633786980962179.0 < x

    1. Initial program 10.4

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

    2. Simplified10.3

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

    3. Taylor expanded around inf 10.6

      \[\leadsto \color{blue}{\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - \left(x + x \cdot \log \left(\frac{1}{x}\right)\right)}\]

    4. Simplified7.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 7633786980962179:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))