Average Error: 0.1 → 0.0
Time: 2.0s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{\sqrt{1 + \sqrt{1 - x \cdot x}}}{\sqrt{x}}\right) + \log \left(\frac{\sqrt{1 + \sqrt{1 - x \cdot x}}}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{\sqrt{1 + \sqrt{1 - x \cdot x}}}{\sqrt{x}}\right) + \log \left(\frac{\sqrt{1 + \sqrt{1 - x \cdot x}}}{\sqrt{x}}\right)
double code(double x) {
	return ((double) log(((double) (((double) (1.0 / x)) + ((double) (((double) sqrt(((double) (1.0 - ((double) (x * x)))))) / x))))));
}
↓
double code(double x) {
	return ((double) (((double) log(((double) (((double) sqrt(((double) (1.0 + ((double) sqrt(((double) (1.0 - ((double) (x * x)))))))))) / ((double) sqrt(x)))))) + ((double) log(((double) (((double) sqrt(((double) (1.0 + ((double) sqrt(((double) (1.0 - ((double) (x * x)))))))))) / ((double) 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. Simplified0.1

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied times-frac0.1

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

  7. Applied log-prod0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))