Average Error: 0.0 → 0.0
Time: 2.7s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{\sqrt{1 - x \cdot x}}\\ \log \left(\frac{1}{x} + t_0 \cdot \frac{t_0}{x}\right) \end{array} \]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\begin{array}{l}
t_0 := \sqrt{\sqrt{1 - x \cdot x}}\\
\log \left(\frac{1}{x} + t_0 \cdot \frac{t_0}{x}\right)
\end{array}

(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt (- 1.0 (* x x))))))
   (log (+ (/ 1.0 x) (* t_0 (/ t_0 x))))))
double code(double x) {
	return log((1.0 / x) + (sqrt(1.0 - (x * x)) / x));
}
↓
double code(double x) {
	double t_0 = sqrt(sqrt(1.0 - (x * x)));
	return log((1.0 / x) + (t_0 * (t_0 / 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. Applied *-un-lft-identity_binary640.0

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

  3. Applied add-sqr-sqrt_binary640.0

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

  4. Applied times-frac_binary640.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022066 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))