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

Average Error: 5.9 → 5.9

Time: 8.9s

Precision: 64

Math

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

↓

\[\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{\sqrt{x}}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt{x}}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

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) {
	return ((double) (((double) (((double) (((double) (((double) (((double) (x - 0.5)) * ((double) (((double) (2.0 * ((double) log(((double) cbrt(x)))))) + ((double) log(((double) cbrt(((double) sqrt(x)))))))))) + ((double) (((double) (x - 0.5)) * ((double) log(((double) cbrt(((double) sqrt(x)))))))))) - x)) + 0.91893853320467)) + ((double) (((double) (((double) (((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)) * z)) + 0.083333333333333)) / x))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.2
Herbie	5.9

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

Derivation

1. Initial program 5.9

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

3. Applied add-cube-cbrt 5.9

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

4. Applied log-prod 5.9

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

5. Applied distribute-lft-in 5.9

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

6. Simplified 5.9

   \[\leadsto \left(\left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
7. Using strategy rm

8. Applied add-sqr-sqrt 5.9

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

9. Applied cbrt-prod 5.9

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

10. Applied log-prod 5.9

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

11. Applied distribute-lft-in 5.9

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

12. Applied associate-+r+ 5.9

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

13. Simplified 5.9

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

14. Final simplification 5.9

    \[\leadsto \left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{\sqrt{x}}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt{x}}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

Reproduce

herbie shell --seed 2020121 
(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)))