Average Error: 0.1 → 0.1
Time: 1.8s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt{\frac{1 + \sqrt{1 - x \cdot x}}{x}}\right) + \log \left(\sqrt{\frac{1 + \sqrt{1 - x \cdot x}}{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\sqrt{\frac{1 + \sqrt{1 - x \cdot x}}{x}}\right) + \log \left(\sqrt{\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 (sqrt (/ (+ 1.0 (sqrt (- 1.0 (* x x)))) x)))
  (log (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) (((double) log(((double) sqrt((((double) (1.0 + ((double) sqrt(((double) (1.0 - ((double) (x * x)))))))) / x))))) + ((double) log(((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 Error: 0.1 bits

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

  2. SimplifiedError: 0.1 bits

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

  4. Applied add-sqr-sqrtError: 0.1 bits

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

  5. Applied log-prodError: 0.1 bits

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

  6. Final simplificationError: 0.1 bits

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

Reproduce

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