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

Average Error: 5.8 → 0.4

Time: 5.2s

Precision: binary64

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 331400.32504989253:\\ \;\;\;\;\left(\left(\sqrt{\log x \cdot \left(x - 0.5\right)} \cdot \sqrt{\log x \cdot \left(x - 0.5\right)} - x\right) + 0.91893853320467\right) + \frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(0.91893853320467 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)\right)\right) + \left(\left(y + 0.0007936500793651\right) \cdot \frac{z}{\frac{x}{z}} - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

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 <= 331400.32504989253)) {
		VAR = ((double) (((double) (((double) (((double) (((double) sqrt(((double) (((double) log(x)) * ((double) (x - 0.5)))))) * ((double) sqrt(((double) (((double) log(x)) * ((double) (x - 0.5)))))))) - x)) + 0.91893853320467)) + (((double) (((double) (z * ((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)))) + 0.083333333333333)) / x)));
	} else {
		VAR = ((double) (((double) (((double) (((double) (x - 0.5)) * ((double) log(((double) sqrt(x)))))) + ((double) (0.91893853320467 + ((double) (((double) (((double) (x - 0.5)) * ((double) log(((double) sqrt(x)))))) - x)))))) + ((double) (((double) (((double) (y + 0.0007936500793651)) * (z / (x / z)))) - ((double) (0.0027777777777778 * (z / x)))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.2
Herbie	0.4

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\]

Derivation

1. Split input into 2 regimes

2. if x < 331400.32504989253

   1. Initial program 0.1

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 0.2

      \[\leadsto \left(\left(\color{blue}{\sqrt{\left(x - 0.5\right) \cdot \log x} \cdot \sqrt{\left(x - 0.5\right) \cdot \log x}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   4. Simplified 0.2

      \[\leadsto \left(\left(\color{blue}{\sqrt{\log x \cdot \left(x - 0.5\right)}} \cdot \sqrt{\left(x - 0.5\right) \cdot \log x} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   5. Simplified 0.2

      \[\leadsto \left(\left(\sqrt{\log x \cdot \left(x - 0.5\right)} \cdot \color{blue}{\sqrt{\log x \cdot \left(x - 0.5\right)}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   if 331400.32504989253 < x

   1. Initial program 10.0

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 10.0

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   4. Applied log-prod 10.0

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{x}\right)\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   5. Applied distribute-lft-in 10.0

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   6. Applied associate--l+ 10.0

      \[\leadsto \left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)\right)} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   7. Applied associate-+l+ 10.0

      \[\leadsto \color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right) + 0.91893853320467\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   8. Simplified 10.0

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \color{blue}{\left(0.91893853320467 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)\right)}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   9. Taylor expanded around inf 10.0

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(0.91893853320467 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)\right)\right) + \color{blue}{\left(\left(0.0007936500793651 \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]

   10. Simplified 0.5

       \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(0.91893853320467 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)\right)\right) + \color{blue}{\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 331400.32504989253:\\ \;\;\;\;\left(\left(\sqrt{\log x \cdot \left(x - 0.5\right)} \cdot \sqrt{\log x \cdot \left(x - 0.5\right)} - x\right) + 0.91893853320467\right) + \frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(0.91893853320467 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)\right)\right) + \left(\left(y + 0.0007936500793651\right) \cdot \frac{z}{\frac{x}{z}} - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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)))