Average Error: 6.0 → 4.2
Time: 27.7s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 52336596543435.78125:\\ \;\;\;\;\frac{\left(\left(x - 0.5\right) \cdot \log x\right) \cdot \left(\left(x - 0.5\right) \cdot \log x\right) - \left(0.9189385332046700050057097541866824030876 - x\right) \cdot \left(0.9189385332046700050057097541866824030876 - x\right)}{\left(x - 0.5\right) \cdot \log x - \left(0.9189385332046700050057097541866824030876 - x\right)} + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]
\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 52336596543435.78125:\\
\;\;\;\;\frac{\left(\left(x - 0.5\right) \cdot \log x\right) \cdot \left(\left(x - 0.5\right) \cdot \log x\right) - \left(0.9189385332046700050057097541866824030876 - x\right) \cdot \left(0.9189385332046700050057097541866824030876 - x\right)}{\left(x - 0.5\right) \cdot \log x - \left(0.9189385332046700050057097541866824030876 - x\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 r318576 = x;
        double r318577 = 0.5;
        double r318578 = r318576 - r318577;
        double r318579 = log(r318576);
        double r318580 = r318578 * r318579;
        double r318581 = r318580 - r318576;
        double r318582 = 0.91893853320467;
        double r318583 = r318581 + r318582;
        double r318584 = y;
        double r318585 = 0.0007936500793651;
        double r318586 = r318584 + r318585;
        double r318587 = z;
        double r318588 = r318586 * r318587;
        double r318589 = 0.0027777777777778;
        double r318590 = r318588 - r318589;
        double r318591 = r318590 * r318587;
        double r318592 = 0.083333333333333;
        double r318593 = r318591 + r318592;
        double r318594 = r318593 / r318576;
        double r318595 = r318583 + r318594;
        return r318595;
}
↓
double f(double x, double y, double z) {
        double r318596 = x;
        double r318597 = 52336596543435.78;
        bool r318598 = r318596 <= r318597;
        double r318599 = 0.5;
        double r318600 = r318596 - r318599;
        double r318601 = log(r318596);
        double r318602 = r318600 * r318601;
        double r318603 = r318602 * r318602;
        double r318604 = 0.91893853320467;
        double r318605 = r318604 - r318596;
        double r318606 = r318605 * r318605;
        double r318607 = r318603 - r318606;
        double r318608 = r318602 - r318605;
        double r318609 = r318607 / r318608;
        double r318610 = y;
        double r318611 = 0.0007936500793651;
        double r318612 = r318610 + r318611;
        double r318613 = z;
        double r318614 = r318612 * r318613;
        double r318615 = 0.0027777777777778;
        double r318616 = r318614 - r318615;
        double r318617 = r318616 * r318613;
        double r318618 = 0.083333333333333;
        double r318619 = r318617 + r318618;
        double r318620 = r318619 / r318596;
        double r318621 = r318609 + r318620;
        double r318622 = r318602 - r318596;
        double r318623 = r318622 + r318604;
        double r318624 = 2.0;
        double r318625 = pow(r318613, r318624);
        double r318626 = r318625 / r318596;
        double r318627 = r318626 * r318612;
        double r318628 = r318613 / r318596;
        double r318629 = r318615 * r318628;
        double r318630 = r318627 - r318629;
        double r318631 = r318623 + r318630;
        double r318632 = r318598 ? r318621 : r318631;
        return r318632;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.0
Target1.4
Herbie4.2
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{0.08333333333333299564049667651488562114537}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 52336596543435.78

    1. Initial program 0.2

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

    3. Applied sub-neg0.2

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

    4. Applied associate-+l+0.2

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

    5. Simplified0.2

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

    7. Applied flip-+0.2

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

    if 52336596543435.78 < x

    1. Initial program 10.6

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

    2. Taylor expanded around inf 10.8

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

    3. Simplified7.5

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

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 52336596543435.78125:\\ \;\;\;\;\frac{\left(\left(x - 0.5\right) \cdot \log x\right) \cdot \left(\left(x - 0.5\right) \cdot \log x\right) - \left(0.9189385332046700050057097541866824030876 - x\right) \cdot \left(0.9189385332046700050057097541866824030876 - x\right)}{\left(x - 0.5\right) \cdot \log x - \left(0.9189385332046700050057097541866824030876 - x\right)} + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))