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

Average Error: 5.8 → 3.0

Time: 7.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 5.24214204453371374 \cdot 10^{122}:\\ \;\;\;\;\left(\left(\frac{\left(x \cdot x - 0.5 \cdot 0.5\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\frac{1}{x}}{\left(0.400000000000006406 \cdot z + 12.000000000000048\right) - 0.100952278095241613 \cdot {z}^{2}}\\ \end{array}\]

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 <= 5.242142044533714e+122)) {
		VAR = (((((((x * x) - (0.5 * 0.5)) * log(x)) / (x + 0.5)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	} else {
		VAR = (((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((1.0 / x) / (((0.4000000000000064 * z) + 12.000000000000048) - (0.10095227809524161 * pow(z, 2.0)))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.1
Herbie	3.0

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

Derivation

1. Split input into 2 regimes

2. if x < 5.242142044533714e+122

   1. Initial program 1.6

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

   3. Applied flip-- 1.6

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

   4. Applied associate-*l/ 1.6

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

   if 5.242142044533714e+122 < x

   1. Initial program 12.8

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

   3. Applied clear-num 12.8

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

   5. Applied div-inv 12.8

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

   6. Applied associate-/r* 12.8

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

   7. Taylor expanded around 0 5.3

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\frac{1}{x}}{\color{blue}{\left(0.400000000000006406 \cdot z + 12.000000000000048\right) - 0.100952278095241613 \cdot {z}^{2}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 5.24214204453371374 \cdot 10^{122}:\\ \;\;\;\;\left(\left(\frac{\left(x \cdot x - 0.5 \cdot 0.5\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\frac{1}{x}}{\left(0.400000000000006406 \cdot z + 12.000000000000048\right) - 0.100952278095241613 \cdot {z}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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)))