Average Error: 0.1 → 0.2
Time: 4.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{\sqrt{x}} \cdot \sqrt[3]{x} + \sqrt{x} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) - \log \left(\sqrt{x} \cdot \sqrt[3]{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{\sqrt{x}} \cdot \sqrt[3]{x} + \sqrt{x} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) - \log \left(\sqrt{x} \cdot \sqrt[3]{x}\right)
double code(double x) {
	return ((double) log(((double) (((double) (1.0 / x)) + ((double) (((double) sqrt(((double) (1.0 - ((double) (x * x)))))) / x))))));
}
↓
double code(double x) {
	return ((double) (((double) log(((double) (((double) (((double) (1.0 / ((double) sqrt(x)))) * ((double) cbrt(x)))) + ((double) (((double) sqrt(x)) * ((double) (((double) sqrt(((double) (1.0 - ((double) (x * x)))))) / ((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))))))) - ((double) log(((double) (((double) sqrt(x)) * ((double) 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 0.1

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

  3. Applied add-cube-cbrt0.1

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

  4. Applied associate-/r*0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied associate-/r*0.1

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

  7. Applied frac-add0.1

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

  8. Applied log-div0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020113 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))