Average Error: 0.0 → 0.0
Time: 1.9s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
↓
\[\log \left(\frac{1 + \sqrt{1 - x \cdot x}}{x}\right) \]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1 + \sqrt{1 - x \cdot x}}{x}\right)

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied add-exp-log_binary640.0

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

  4. Simplified0.0

    \[\leadsto \log \left(e^{\color{blue}{\log \left(\frac{1 + \sqrt{1 - x \cdot x}}{x}\right)}}\right) \]
  5. Using strategy rm

  6. Applied rem-exp-log_binary640.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2021202 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))