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

Average Error: 5.8 → 3.6

Time: 14.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z = -inf.0 \lor \neg \left(\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 1.85056140285941314 \cdot 10^{272}\right):\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \log \left({\left(e^{\frac{1}{x}}\right)}^{\left(\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{1}{\frac{x}{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}}\\ \end{array}\]

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) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)) * z)) + 0.083333333333333)) / x))));
}

↓

double code(double x, double y, double z) {
	double VAR;
	if (((((double) (((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)) * z)) <= -inf.0) || !(((double) (((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)) * z)) <= 1.850561402859413e+272))) {
		VAR = ((double) (((double) (((double) (((double) (((double) (x - 0.5)) * ((double) log(x)))) - x)) + 0.91893853320467)) + ((double) log(((double) pow(((double) exp(((double) (1.0 / x)))), ((double) (((double) (((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)) * z)) + 0.083333333333333))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (((double) (x - 0.5)) * ((double) log(x)))) - x)) + 0.91893853320467)) + ((double) (1.0 / ((double) (x / ((double) (((double) (((double) (((double) (((double) (y + 0.0007936500793651)) * z)) - 0.0027777777777778)) * z)) + 0.083333333333333))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.1
Herbie	3.6

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

Derivation

1. Split input into 2 regimes

2. if (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < -inf.0 or 1.850561402859413e+272 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)

   1. Initial program 53.6

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

   3. Applied clear-num 53.6

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

   5. Applied div-inv 53.6

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

   6. Applied associate-/r* 53.6

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

   8. Applied add-log-exp 63.7

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

   9. Simplified 31.2

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

   if -inf.0 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < 1.850561402859413e+272

   1. Initial program 0.2

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

   3. Applied clear-num 0.3

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \color{blue}{\frac{1}{\frac{x}{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z = -inf.0 \lor \neg \left(\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 1.85056140285941314 \cdot 10^{272}\right):\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \log \left({\left(e^{\frac{1}{x}}\right)}^{\left(\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{1}{\frac{x}{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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)))