Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Error: 5.8 → 0.4

Time: 6.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -306537321.28306544:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \left(\left(z \cdot \frac{z}{x}\right) \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - \frac{z}{x} \cdot 0.0027777777777778\right)\\ \mathbf{elif}\;z \le 145638950479.37927:\\ \;\;\;\;\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 + \left(\frac{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{z} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\right) + 0.0833333333333329956}{x} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467001 + \left(\left(x - 0.5\right) \cdot \log x - x\right)\right) + \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot \frac{z}{\frac{x}{z}} - \frac{z}{x} \cdot 0.0027777777777778\right)\\ \end{array}\]

C

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 ((z <= -306537321.28306544)) {
		VAR = ((double) (((double) (((double) (((double) (((double) (((double) (x - 0.5)) * ((double) (2.0 * ((double) log(((double) cbrt(x)))))))) + ((double) (((double) (x - 0.5)) * ((double) log(((double) cbrt(x)))))))) - x)) + 0.91893853320467)) + ((double) (((double) (((double) (z * ((double) (z / x)))) * ((double) (y + 0.0007936500793651)))) - ((double) (((double) (z / x)) * 0.0027777777777778))))));
	} else {
		double VAR_1;
		if ((z <= 145638950479.37927)) {
			VAR_1 = ((double) (((double) (((double) (x - 0.5)) * ((double) log(x)))) + ((double) (0.91893853320467 + ((double) (((double) (((double) (((double) (((double) (((double) cbrt(z)) * ((double) cbrt(z)))) * ((double) (((double) cbrt(z)) * ((double) (((double) (z * ((double) (y + 0.0007936500793651)))) - 0.0027777777777778)))))) + 0.083333333333333)) / x)) - x))))));
		} else {
			VAR_1 = ((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) (((double) (z / x)) * 0.0027777777777778))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.1
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if z < -306537321.28306544

   1. Initial program 20.1

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

   3. Applied add-cube-cbrt 20.1

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

   4. Applied log-prod 20.2

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

   5. Applied distribute-lft-in 20.1

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

   6. Simplified 20.1

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

   7. Taylor expanded around inf 20.6

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

   8. Simplified 0.8

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

   if -306537321.28306544 < z < 145638950479.37927

   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. Simplified 0.2

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

   4. Applied add-cube-cbrt 0.3

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

   5. Applied associate-*l* 0.3

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

   6. Simplified 0.3

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

   if 145638950479.37927 < z

   1. Initial program 21.0

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

   2. Taylor expanded around inf 21.4

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

   3. Simplified 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -306537321.28306544:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \left(\left(z \cdot \frac{z}{x}\right) \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - \frac{z}{x} \cdot 0.0027777777777778\right)\\ \mathbf{elif}\;z \le 145638950479.37927:\\ \;\;\;\;\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 + \left(\frac{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{z} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\right) + 0.0833333333333329956}{x} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467001 + \left(\left(x - 0.5\right) \cdot \log x - x\right)\right) + \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot \frac{z}{\frac{x}{z}} - \frac{z}{x} \cdot 0.0027777777777778\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))