Average Error: 0.1 → 0.1
Time: 2.5s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\frac{\sqrt{1 - x \cdot x}}{\sqrt{x}}}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\frac{\sqrt{1 - x \cdot x}}{\sqrt{x}}}{\sqrt{x}}\right)
(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))
↓
(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (/ (sqrt (- 1.0 (* x x))) (sqrt x)) (sqrt x)))))
double code(double x) {
	return log((1.0 / x) + (sqrt(1.0 - (x * x)) / x));
}
↓
double code(double x) {
	return log((1.0 / x) + ((sqrt(1.0 - (x * x)) / sqrt(x)) / sqrt(x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt_binary640.1

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

  4. Applied associate-/r*_binary640.1

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

  5. Final simplification0.1

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

Alternatives

Reproduce

herbie shell --seed 2021113 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))