Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B

Average Error: 0.0 → 5.3

Time: 2.6s

Precision: 64

Math

\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.52295109951887609 \cdot 10^{-17} \lor \neg \left(x \le 6.68360706888572707 \cdot 10^{-37}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{y}{\left(-{\left(\frac{x \cdot y}{2}\right)}^{3} \cdot \frac{x \cdot y}{2}\right) + {1}^{4}} \cdot \left(1 \cdot 1 + \frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}\right)\right) \cdot \left(1 - \frac{x \cdot y}{2}\right)\\ \end{array}\]

C

double code(double x, double y) {
	return (x - (y / (1.0 + ((x * y) / 2.0))));
}

↓

double code(double x, double y) {
	double VAR;
	if (((x <= -2.522951099518876e-17) || !(x <= 6.683607068885727e-37))) {
		VAR = (x - (2.0 / x));
	} else {
		VAR = (x - (((y / (-(pow(((x * y) / 2.0), 3.0) * ((x * y) / 2.0)) + pow(1.0, 4.0))) * ((1.0 * 1.0) + (((x * y) / 2.0) * ((x * y) / 2.0)))) * (1.0 - ((x * y) / 2.0))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if x < -2.522951099518876e-17 or 6.683607068885727e-37 < x

   1. Initial program 0.0

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]

   2. Taylor expanded around inf 3.0

      \[\leadsto \color{blue}{x - 2 \cdot \frac{1}{x}}\]

   3. Simplified 3.0

      \[\leadsto \color{blue}{x - \frac{2}{x}}\]

   if -2.522951099518876e-17 < x < 6.683607068885727e-37

   1. Initial program 0.1

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
   2. Using strategy rm

   3. Applied flip-+ 3.0

      \[\leadsto x - \frac{y}{\color{blue}{\frac{1 \cdot 1 - \frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}}{1 - \frac{x \cdot y}{2}}}}\]

   4. Applied associate-/r/ 3.1

      \[\leadsto x - \color{blue}{\frac{y}{1 \cdot 1 - \frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}} \cdot \left(1 - \frac{x \cdot y}{2}\right)}\]
   5. Using strategy rm

   6. Applied flip-- 8.1

      \[\leadsto x - \frac{y}{\color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}\right) \cdot \left(\frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}\right)}{1 \cdot 1 + \frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}}}} \cdot \left(1 - \frac{x \cdot y}{2}\right)\]

   7. Applied associate-/r/ 8.2

      \[\leadsto x - \color{blue}{\left(\frac{y}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}\right) \cdot \left(\frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}\right)} \cdot \left(1 \cdot 1 + \frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}\right)\right)} \cdot \left(1 - \frac{x \cdot y}{2}\right)\]

   8. Simplified 8.2

      \[\leadsto x - \left(\color{blue}{\frac{y}{\left(-{\left(\frac{x \cdot y}{2}\right)}^{3} \cdot \frac{x \cdot y}{2}\right) + {1}^{4}}} \cdot \left(1 \cdot 1 + \frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}\right)\right) \cdot \left(1 - \frac{x \cdot y}{2}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.52295109951887609 \cdot 10^{-17} \lor \neg \left(x \le 6.68360706888572707 \cdot 10^{-37}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{y}{\left(-{\left(\frac{x \cdot y}{2}\right)}^{3} \cdot \frac{x \cdot y}{2}\right) + {1}^{4}} \cdot \left(1 \cdot 1 + \frac{x \cdot y}{2} \cdot \frac{x \cdot y}{2}\right)\right) \cdot \left(1 - \frac{x \cdot y}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1 (/ (* x y) 2)))))