Average Error: 0.1 → 0.2
Time: 4.0s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\frac{1}{\sqrt[3]{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\frac{1}{\sqrt[3]{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)
double code(double x) {
	return log(((1.0 / x) + (sqrt((1.0 - (x * x))) / x)));
}
↓
double code(double x) {
	return (log((1.0 / (cbrt(x) * cbrt(x)))) + log(((1.0 / cbrt(x)) + (sqrt((1.0 - (x * x))) / 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 *-un-lft-identity0.1

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

  5. Applied times-frac0.1

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

  6. Applied add-cube-cbrt0.1

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

  7. Applied *-un-lft-identity0.1

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

  8. Applied times-frac0.1

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

  9. Applied distribute-lft-out0.1

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

  10. Applied log-prod0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))