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

Average Error: 6.4 → 1.3

Time: 5.9s

Precision: binary64

Math

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

↓

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	1.3
Herbie	1.3

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

Derivation

1. Initial program 6.4

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

2. Simplified 6.4

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

4. Applied sub-neg 6.4

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

5. Applied distribute-lft-in 6.4

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

6. Simplified 6.4

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

8. Applied distribute-rgt-neg-out 6.4

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

9. Applied unsub-neg 6.4

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

10. Applied associate-+l- 6.4

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

11. Applied div-sub 6.4

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

12. Simplified 1.3

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

13. Final simplification 1.3

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

Reproduce

herbie shell --seed 2020182 
(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)))