Average Error: 0.1 → 0.1
Time: 1.3s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{\sqrt{1 + \sqrt{1 - x \cdot x}} \cdot \sqrt{1 + \sqrt{1 - x \cdot x}}}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{\sqrt{1 + \sqrt{1 - x \cdot x}} \cdot \sqrt{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
  (/
   (*
    (sqrt (+ 1.0 (sqrt (- 1.0 (* x x)))))
    (sqrt (+ 1.0 (sqrt (- 1.0 (* x x))))))
   x)))
double code(double x) {
	return ((double) log(((double) ((1.0 / x) + (((double) sqrt(((double) (1.0 - ((double) (x * x)))))) / x)))));
}
↓
double code(double x) {
	return ((double) log((((double) (((double) sqrt(((double) (1.0 + ((double) sqrt(((double) (1.0 - ((double) (x * x)))))))))) * ((double) sqrt(((double) (1.0 + ((double) sqrt(((double) (1.0 - ((double) (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 0.1

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

  2. Simplified0.1

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

  4. Applied add-sqr-sqrt_binary640.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020205 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))