Average Error: 0.0 → 0.0
Time: 10.8s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\left|\sqrt[3]{1 - x \cdot x}\right|}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{\sqrt[3]{1 - x \cdot x}}}{\sqrt[3]{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\left|\sqrt[3]{1 - x \cdot x}\right|}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{\sqrt[3]{1 - x \cdot x}}}{\sqrt[3]{x}}\right)
double f(double x) {
        double r73957 = 1.0;
        double r73958 = x;
        double r73959 = r73957 / r73958;
        double r73960 = r73958 * r73958;
        double r73961 = r73957 - r73960;
        double r73962 = sqrt(r73961);
        double r73963 = r73962 / r73958;
        double r73964 = r73959 + r73963;
        double r73965 = log(r73964);
        return r73965;
}
↓
double f(double x) {
        double r73966 = 1.0;
        double r73967 = x;
        double r73968 = r73966 / r73967;
        double r73969 = r73967 * r73967;
        double r73970 = r73966 - r73969;
        double r73971 = cbrt(r73970);
        double r73972 = fabs(r73971);
        double r73973 = cbrt(r73967);
        double r73974 = r73973 * r73973;
        double r73975 = r73972 / r73974;
        double r73976 = sqrt(r73971);
        double r73977 = r73976 / r73973;
        double r73978 = r73975 * r73977;
        double r73979 = r73968 + r73978;
        double r73980 = log(r73979);
        return r73980;
}

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-cube-cbrt0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied sqrt-prod0.0

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

  6. Applied times-frac0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))