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

Average Error: 5.9 → 0.7

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \le -1.35166735028178523 \cdot 10^{-13} \lor \neg \left(z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \le 3.5269674065843528 \cdot 10^{274}\right):\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot \left(z \cdot \frac{z}{x}\right) - z \cdot \frac{0.0027777777777778}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467001 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left(\left|\sqrt[3]{x}\right|\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right) - x\right)\right)\right) + \frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x}\\ \end{array}\]

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 (((((double) (z * ((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)))) <= -1.3516673502817852e-13) || !(((double) (z * ((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)))) <= 3.526967406584353e+274))) {
		VAR = ((double) (((double) (((double) (((double) (((double) (x - 0.5)) * ((double) log(x)))) - x)) + 0.91893853320467)) + ((double) (((double) (((double) (y + 0.0007936500793651)) * ((double) (z * ((double) (z / x)))))) - ((double) (z * ((double) (0.0027777777777778 / x))))))));
	} else {
		VAR = ((double) (((double) (0.91893853320467 + ((double) (((double) (((double) (x - 0.5)) * ((double) log(((double) sqrt(x)))))) + ((double) (((double) (((double) (((double) (x - 0.5)) * ((double) log(((double) fabs(((double) cbrt(x)))))))) + ((double) (((double) (x - 0.5)) * ((double) log(((double) sqrt(((double) cbrt(x)))))))))) - x)))))) + ((double) (((double) (((double) (z * ((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)))) + 0.083333333333333)) / x))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.1
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < -1.35166735028178523e-13 or 3.5269674065843528e274 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)

   1. Initial program 31.2

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

   2. Taylor expanded around inf 33.0

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

   3. Simplified 3.0

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

   if -1.35166735028178523e-13 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < 3.5269674065843528e274

   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-sqr-sqrt 0.2

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

   4. Applied log-prod 0.2

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

   5. Applied distribute-lft-in 0.2

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

   6. Applied associate--l+ 0.3

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

   8. Applied add-cube-cbrt 0.3

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

   9. Applied sqrt-prod 0.3

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

   10. Applied log-prod 0.2

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

   11. Applied distribute-lft-in 0.2

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

   12. Simplified 0.2

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \le -1.35166735028178523 \cdot 10^{-13} \lor \neg \left(z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \le 3.5269674065843528 \cdot 10^{274}\right):\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot \left(z \cdot \frac{z}{x}\right) - z \cdot \frac{0.0027777777777778}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467001 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left(\left|\sqrt[3]{x}\right|\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right) - x\right)\right)\right) + \frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x}\\ \end{array}\]

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)))