Average Error: 0.0 → 0.0
Time: 4.1s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
↓
\[\log \left(\frac{1}{x} + \frac{\frac{\sqrt{1 - x \cdot x}}{{\left(\sqrt[3]{x}\right)}^{2}}}{\sqrt[3]{x}}\right) \]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\frac{\sqrt{1 - x \cdot x}}{{\left(\sqrt[3]{x}\right)}^{2}}}{\sqrt[3]{x}}\right)

(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))
↓
(FPCore (x)
 :precision binary64
 (log
  (+ (/ 1.0 x) (/ (/ (sqrt (- 1.0 (* x x))) (pow (cbrt x) 2.0)) (cbrt 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) + ((sqrt(1.0 - (x * x)) / pow(cbrt(x), 2.0)) / cbrt(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 add-cube-cbrt_binary640.0

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

  3. Applied associate-/r*_binary640.0

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

  4. Applied pow2_binary640.0

    \[\leadsto \log \left(\frac{1}{x} + \frac{\frac{\sqrt{1 - x \cdot x}}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}}{\sqrt[3]{x}}\right) \]

  5. Final simplification0.0

    \[\leadsto \log \left(\frac{1}{x} + \frac{\frac{\sqrt{1 - x \cdot x}}{{\left(\sqrt[3]{x}\right)}^{2}}}{\sqrt[3]{x}}\right) \]

Reproduce

herbie shell --seed 2022082 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))