Average Error: 0.0 → 0.0
Time: 2.4s
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-log-exp_binary64_79963.5

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

  4. Applied add-log-exp_binary64_79963.5

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

  5. Applied sum-log_binary64_85163.6

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

  6. Simplified63.6

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

  8. Applied rem-log-exp_binary64_8010.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2021058 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))