Average Error: 0.0 → 0.0
Time: 6.0s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \left|\sqrt[3]{1 - x \cdot x}\right| \cdot \frac{\sqrt{\sqrt[3]{1 - x \cdot x}}}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \left|\sqrt[3]{1 - x \cdot x}\right| \cdot \frac{\sqrt{\sqrt[3]{1 - x \cdot x}}}{x}\right)
double f(double x) {
        double r76521 = 1.0;
        double r76522 = x;
        double r76523 = r76521 / r76522;
        double r76524 = r76522 * r76522;
        double r76525 = r76521 - r76524;
        double r76526 = sqrt(r76525);
        double r76527 = r76526 / r76522;
        double r76528 = r76523 + r76527;
        double r76529 = log(r76528);
        return r76529;
}
↓
double f(double x) {
        double r76530 = 1.0;
        double r76531 = x;
        double r76532 = r76530 / r76531;
        double r76533 = r76531 * r76531;
        double r76534 = r76530 - r76533;
        double r76535 = cbrt(r76534);
        double r76536 = fabs(r76535);
        double r76537 = sqrt(r76535);
        double r76538 = r76537 / r76531;
        double r76539 = r76536 * r76538;
        double r76540 = r76532 + r76539;
        double r76541 = log(r76540);
        return r76541;
}

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 *-un-lft-identity0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied sqrt-prod0.0

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

  6. Applied times-frac0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))