Average Error: 0.0 → 0.0
Time: 5.3s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\left|\sqrt[3]{1 - x \cdot x}\right|}{\sqrt{x} \cdot 1} \cdot \frac{\sqrt{\sqrt[3]{1 - x \cdot x}}}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\left|\sqrt[3]{1 - x \cdot x}\right|}{\sqrt{x} \cdot 1} \cdot \frac{\sqrt{\sqrt[3]{1 - x \cdot x}}}{\sqrt{x}}\right)
double f(double x) {
        double r94445 = 1.0;
        double r94446 = x;
        double r94447 = r94445 / r94446;
        double r94448 = r94446 * r94446;
        double r94449 = r94445 - r94448;
        double r94450 = sqrt(r94449);
        double r94451 = r94450 / r94446;
        double r94452 = r94447 + r94451;
        double r94453 = log(r94452);
        return r94453;
}
↓
double f(double x) {
        double r94454 = 1.0;
        double r94455 = x;
        double r94456 = r94454 / r94455;
        double r94457 = r94455 * r94455;
        double r94458 = r94454 - r94457;
        double r94459 = cbrt(r94458);
        double r94460 = fabs(r94459);
        double r94461 = sqrt(r94455);
        double r94462 = 1.0;
        double r94463 = r94461 * r94462;
        double r94464 = r94460 / r94463;
        double r94465 = sqrt(r94459);
        double r94466 = r94465 / r94461;
        double r94467 = r94464 * r94466;
        double r94468 = r94456 + r94467;
        double r94469 = log(r94468);
        return r94469;
}

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 add-sqr-sqrt0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied sqrt-prod0.0

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

  6. Applied times-frac0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))