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

Average Error: 6.1 → 1.0

Time: 22.4min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 64580610598537.156:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{elif}\;x \le 2.4155518908026999 \cdot 10^{197}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{z}{x} \cdot \left(z \cdot \left(7.93650079365100015 \cdot 10^{-4} + y\right) - 0.0027777777777778\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \left(\frac{0.0833333333333329956}{x} - \frac{z}{x} \cdot \left(0.0027777777777778 - 7.93650079365100015 \cdot 10^{-4} \cdot z\right)\right)\\ \end{array}\]

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) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)) * z)) + 0.083333333333333)) / x)));
}

↓

double code(double x, double y, double z) {
	double VAR;
	if ((x <= 64580610598537.16)) {
		VAR = ((double) (((double) (((double) (((double) (x - 0.5)) * ((double) log(x)))) + ((double) (0.91893853320467 - x)))) + (((double) (((double) (((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)) * z)) + 0.083333333333333)) / x)));
	} else {
		double VAR_1;
		if ((x <= 2.4155518908027e+197)) {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (x - 0.5)) * ((double) log(x)))) - x)) + 0.91893853320467)) + ((double) ((z / x) * ((double) (((double) (z * ((double) (0.0007936500793651 + y)))) - 0.0027777777777778))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (((double) (2.0 * ((double) log(((double) cbrt(x)))))) * ((double) (x - 0.5)))) + ((double) (((double) (x - 0.5)) * ((double) log(((double) cbrt(x)))))))) - x)) + 0.91893853320467)) + ((double) ((0.083333333333333 / x) - ((double) ((z / x) * ((double) (0.0027777777777778 - ((double) (0.0007936500793651 * z))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	1.3
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if x < 64580610598537.156

   1. Initial program 0.1

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

   3. Applied sub-neg 0.1

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

   4. Applied associate-+l+ 0.1

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

   5. Simplified 0.1

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

   if 64580610598537.156 < x < 2.4155518908026999e197

   1. Initial program 7.3

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

   2. Taylor expanded around inf 7.5

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

   3. Simplified 1.3

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

   if 2.4155518908026999e197 < x

   1. Initial program 15.4

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

   2. Taylor expanded around 0 11.5

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \color{blue}{\left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]

   3. Simplified 2.2

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

   5. Applied add-cube-cbrt 2.2

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

   6. Applied log-prod 2.3

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

   7. Applied distribute-lft-in 2.3

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

   8. Simplified 2.3

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 64580610598537.156:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{elif}\;x \le 2.4155518908026999 \cdot 10^{197}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{z}{x} \cdot \left(z \cdot \left(7.93650079365100015 \cdot 10^{-4} + y\right) - 0.0027777777777778\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \left(\frac{0.0833333333333329956}{x} - \frac{z}{x} \cdot \left(0.0027777777777778 - 7.93650079365100015 \cdot 10^{-4} \cdot z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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)))