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

Average Error: 5.9 → 4.3

Time: 7.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 2.2940227510868728 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) + \left(\sqrt{\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right)} \cdot \sqrt{\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right)} - x\right)\right) + 0.91893853320467\right) + \frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) - x\right)\right)\right) + \left(\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x} - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.2940227510868728e+23)
   (+
    (+
     (+
      (* (log (sqrt x)) (- x 0.5))
      (-
       (*
        (sqrt (* (log (sqrt x)) (- x 0.5)))
        (sqrt (* (log (sqrt x)) (- x 0.5))))
       x))
     0.91893853320467)
    (/
     (+
      (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
      0.083333333333333)
     x))
   (+
    (+
     0.91893853320467
     (+ (* (log (sqrt x)) (- x 0.5)) (- (* (log (sqrt x)) (- x 0.5)) x)))
    (-
     (* (+ y 0.0007936500793651) (/ (* z z) x))
     (* 0.0027777777777778 (/ z x))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.2940227510868728e+23) {
		tmp = (((log(sqrt(x)) * (x - 0.5)) + ((sqrt(log(sqrt(x)) * (x - 0.5)) * sqrt(log(sqrt(x)) * (x - 0.5))) - x)) + 0.91893853320467) + (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x);
	} else {
		tmp = (0.91893853320467 + ((log(sqrt(x)) * (x - 0.5)) + ((log(sqrt(x)) * (x - 0.5)) - x))) + (((y + 0.0007936500793651) * ((z * z) / x)) - (0.0027777777777778 * (z / x)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.2
Herbie	4.3

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2940227510868728e23

   1. Initial program 0.2

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

   3. Applied add-sqr-sqrt_binary64_14252 0.2

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

   4. Applied log-prod_binary64_14314 0.2

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

   5. Applied distribute-rgt-in_binary64_14183 0.2

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

   6. Applied associate--l+_binary64_14170 0.2

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

   7. Simplified 0.2

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

   9. Applied add-sqr-sqrt_binary64_14252 0.3

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

   if 2.2940227510868728e23 < x

   1. Initial program 10.6

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

   3. Applied add-sqr-sqrt_binary64_14252 10.6

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

   4. Applied log-prod_binary64_14314 10.6

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

   5. Applied distribute-rgt-in_binary64_14183 10.6

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

   6. Applied associate--l+_binary64_14170 10.6

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

   7. Simplified 10.6

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

   8. Taylor expanded around inf 10.8

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

   9. Simplified 7.6

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

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2940227510868728 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) + \left(\sqrt{\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right)} \cdot \sqrt{\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right)} - x\right)\right) + 0.91893853320467\right) + \frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) - x\right)\right)\right) + \left(\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x} - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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)))