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

Average Error: 5.7 → 0.3

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 70802251299.52406:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(0.91893853320467 + \left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(0.91893853320467 - x\right)\right)\right)\\ \end{array}\]

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 <= 70802251299.52406)) {
		VAR = ((double) (((double) (((double) (((double) (x - 0.5)) * ((double) (2.0 * ((double) log(((double) cbrt(x)))))))) + ((double) (((double) (x - 0.5)) * ((double) log(((double) cbrt(x)))))))) + ((double) (0.91893853320467 + ((double) ((((double) (((double) (z * ((double) (((double) (z * ((double) (y + 0.0007936500793651)))) - 0.0027777777777778)))) + 0.083333333333333)) / x) - x))))));
	} else {
		VAR = ((double) (((double) (((double) (x - 0.5)) * ((double) log(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))) + ((double) (((double) (((double) (((double) (y + 0.0007936500793651)) * ((double) (z * (z / x))))) - ((double) (0.0027777777777778 * (z / x))))) + ((double) (((double) (((double) (x - 0.5)) * ((double) log(((double) cbrt(x)))))) + ((double) (0.91893853320467 - x))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.7
Target	1.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < 70802251299.5240631

   1. Initial program Error: 0.1 bits

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

   2. Simplified Error: 0.1 bits

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

   4. Applied add-cube-cbrt Error: 0.2 bits

      \[\leadsto \left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} + \left(0.91893853320467 + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\right)\right)\]

   5. Applied log-prod Error: 0.2 bits

      \[\leadsto \left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} + \left(0.91893853320467 + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\right)\right)\]

   6. Applied distribute-lft-in Error: 0.2 bits

      \[\leadsto \color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(0.91893853320467 + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\right)\right)\]

   7. Simplified Error: 0.2 bits

      \[\leadsto \left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(0.91893853320467 + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\right)\right)\]

   if 70802251299.5240631 < x

   1. Initial program Error: 9.9 bits

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

   2. Simplified Error: 9.9 bits

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

   4. Applied add-cube-cbrt Error: 9.9 bits

      \[\leadsto \left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} + \left(0.91893853320467 + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\right)\right)\]

   5. Applied log-prod Error: 9.9 bits

      \[\leadsto \left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} + \left(0.91893853320467 + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\right)\right)\]

   6. Applied distribute-lft-in Error: 9.9 bits

      \[\leadsto \color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(0.91893853320467 + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\right)\right)\]

   7. Applied associate-+l+ Error: 9.9 bits

      \[\leadsto \color{blue}{\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(0.91893853320467 + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\right)\right)\right)}\]

   8. Simplified Error: 9.9 bits

      \[\leadsto \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(\left(0.91893853320467 - x\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)\right)}\]

   9. Taylor expanded around inf Error: 9.9 bits

      \[\leadsto \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\color{blue}{\left(\left(0.0007936500793651 \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)} + \left(\left(0.91893853320467 - x\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)\right)\]

   10. Simplified Error: 0.5 bits

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

4. Final simplification Error: 0.3 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 70802251299.52406:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(0.91893853320467 + \left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(0.91893853320467 - x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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)))