Average Error: 0.1 → 0.1
Time: 5.0s
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 0.1

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

  3. Applied frac-add_binary64_179130.9

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

  4. Simplified30.9

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

  6. Applied associate-/r*_binary64_17270.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2021093 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))