Average Error: 0.0 → 0.4
Time: 1.6s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
↓
\[\log \left(\frac{1}{x} + \left(\frac{1}{x} - x \cdot 0.5\right)\right) \]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \left(\frac{1}{x} - x \cdot 0.5\right)\right)

(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))
↓
(FPCore (x) :precision binary64 (log (+ (/ 1.0 x) (- (/ 1.0 x) (* x 0.5)))))
double code(double x) {
	return log((1.0 / x) + (sqrt(1.0 - (x * x)) / x));
}
↓
double code(double x) {
	return log((1.0 / x) + ((1.0 / x) - (x * 0.5)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]

  2. Taylor expanded around 0 0.4

    \[\leadsto \log \left(\frac{1}{x} + \color{blue}{\left(\frac{1}{x} - 0.5 \cdot x\right)}\right) \]

  3. Simplified0.4

    \[\leadsto \log \left(\frac{1}{x} + \color{blue}{\left(\frac{1}{x} - x \cdot 0.5\right)}\right) \]

  4. Final simplification0.4

    \[\leadsto \log \left(\frac{1}{x} + \left(\frac{1}{x} - x \cdot 0.5\right)\right) \]

Reproduce

herbie shell --seed 2021196 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))