Average Error: 0.0 → 0.0
Time: 3.3s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \left(\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \sqrt[3]{\sqrt{1 - x \cdot x}}\right)}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \left(\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \sqrt[3]{\sqrt{1 - x \cdot x}}\right)}{x}\right)
(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))
↓
(FPCore (x)
 :precision binary64
 (log
  (+
   (/ 1.0 x)
   (/
    (*
     (cbrt (sqrt (- 1.0 (* x x))))
     (* (cbrt (sqrt (- 1.0 (* x x)))) (cbrt (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 / x) + ((cbrt(sqrt(1.0 - (x * x))) * (cbrt(sqrt(1.0 - (x * x))) * cbrt(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-cube-cbrt_binary640.0

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

  4. Final simplification0.0

    \[\leadsto \log \left(\frac{1}{x} + \frac{\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \left(\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \sqrt[3]{\sqrt{1 - x \cdot x}}\right)}{x}\right)\]

Reproduce

herbie shell --seed 2021139 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))