Average Error: 6.0 → 0.4
Time: 6.7s
Precision: binary64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 16073734415565289500:\\ \;\;\;\;\left(\left(\log \left(\sqrt[3]{-1} \cdot \sqrt[3]{\frac{1}{\frac{-1}{x}}}\right) \cdot \left(x \cdot 3 - 1.5\right) - x\right) + 0.91893853320467001\right) + \frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467001 + \left(\left(x - 0.5\right) \cdot \log x - x\right)\right) + \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot \frac{z}{\frac{x}{z}} - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]
\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 16073734415565289500:\\
\;\;\;\;\left(\left(\log \left(\sqrt[3]{-1} \cdot \sqrt[3]{\frac{1}{\frac{-1}{x}}}\right) \cdot \left(x \cdot 3 - 1.5\right) - x\right) + 0.91893853320467001\right) + \frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x}\\

\mathbf{else}:\\
\;\;\;\;\left(0.91893853320467001 + \left(\left(x - 0.5\right) \cdot \log x - x\right)\right) + \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot \frac{z}{\frac{x}{z}} - 0.0027777777777778 \cdot \frac{z}{x}\right)\\

\end{array}
double code(double x, double y, double z) {
	return ((double) (((double) (((double) (((double) (((double) (x - 0.5)) * ((double) log(x)))) - x)) + 0.91893853320467)) + ((double) (((double) (((double) (((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)) * z)) + 0.083333333333333)) / x))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((x <= 1.607373441556529e+19)) {
		VAR = ((double) (((double) (((double) (((double) (((double) log(((double) (((double) cbrt(-1.0)) * ((double) cbrt(((double) (1.0 / ((double) (-1.0 / x)))))))))) * ((double) (((double) (x * 3.0)) - 1.5)))) - x)) + 0.91893853320467)) + ((double) (((double) (((double) (z * ((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)))) + 0.083333333333333)) / x))));
	} else {
		VAR = ((double) (((double) (0.91893853320467 + ((double) (((double) (((double) (x - 0.5)) * ((double) log(x)))) - x)))) + ((double) (((double) (((double) (y + 0.0007936500793651)) * ((double) (z / ((double) (x / z)))))) - ((double) (0.0027777777777778 * ((double) (z / x))))))));
	}
	return VAR;
}

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
Target1.1
Herbie0.4
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 16073734415565289500

    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-cube-cbrt0.2

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

    4. Applied log-prod0.2

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

    5. Applied distribute-lft-in0.2

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

    6. Simplified0.2

      \[\leadsto \left(\left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    7. Taylor expanded around -inf 64.0

      \[\leadsto \left(\left(\color{blue}{\left(3 \cdot \left(x \cdot \log \left(\sqrt[3]{-1} \cdot e^{\frac{1}{3} \cdot \left(\log 1 - \log \left(\frac{-1}{x}\right)\right)}\right)\right) - 1.5 \cdot \log \left(\sqrt[3]{-1} \cdot e^{\frac{1}{3} \cdot \left(\log 1 - \log \left(\frac{-1}{x}\right)\right)}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    8. Simplified0.2

      \[\leadsto \left(\left(\color{blue}{\log \left(\sqrt[3]{-1} \cdot \sqrt[3]{\frac{1}{\frac{-1}{x}}}\right) \cdot \left(x \cdot 3 - 1.5\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    if 16073734415565289500 < x

    1. Initial program 10.5

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

    2. Taylor expanded around inf 10.6

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \color{blue}{\left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]

    3. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 16073734415565289500:\\ \;\;\;\;\left(\left(\log \left(\sqrt[3]{-1} \cdot \sqrt[3]{\frac{1}{\frac{-1}{x}}}\right) \cdot \left(x \cdot 3 - 1.5\right) - x\right) + 0.91893853320467001\right) + \frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467001 + \left(\left(x - 0.5\right) \cdot \log x - x\right)\right) + \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot \frac{z}{\frac{x}{z}} - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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)))