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

Average Error: 6.0 → 4.4

Time: 7.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 7.8191391361236409 \cdot 10^{26}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log \left(\sqrt{x}\right), \mathsf{fma}\left(\log \left(\sqrt{x}\right), x - 0.5, 0.91893853320467001 - x\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z) {
	return (((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
}

↓

double code(double x, double y, double z) {
	double VAR;
	if ((x <= 7.819139136123641e+26)) {
		VAR = (fma((x - 0.5), log(sqrt(x)), fma(log(sqrt(x)), (x - 0.5), (0.91893853320467 - x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	} else {
		VAR = fma((pow(z, 2.0) / x), y, ((0.0007936500793651 * (pow(z, 2.0) / x)) - fma(log((1.0 / x)), x, x)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	1.1
Herbie	4.4

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

Derivation

1. Split input into 2 regimes

2. if x < 7.819139136123641e+26

   1. Initial program 0.3

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

   3. Applied add-sqr-sqrt 0.3

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

   4. Applied log-prod 0.3

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

   5. Applied distribute-lft-in 0.3

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

   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. Applied associate-+l+ 0.3

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

   8. Simplified 0.3

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

   10. Applied fma-def 0.2

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

   if 7.819139136123641e+26 < x

   1. Initial program 10.8

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

   2. Simplified 10.7

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

   3. Taylor expanded around inf 10.9

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

   4. Simplified 7.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 7.8191391361236409 \cdot 10^{26}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log \left(\sqrt{x}\right), \mathsf{fma}\left(\log \left(\sqrt{x}\right), x - 0.5, 0.91893853320467001 - x\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(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)))