Average Error: 0.0 → 0.0
Time: 13.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)
double f(double x) {
        double r1141012 = 1.0;
        double r1141013 = x;
        double r1141014 = r1141012 / r1141013;
        double r1141015 = r1141013 * r1141013;
        double r1141016 = r1141012 - r1141015;
        double r1141017 = sqrt(r1141016);
        double r1141018 = r1141017 / r1141013;
        double r1141019 = r1141014 + r1141018;
        double r1141020 = log(r1141019);
        return r1141020;
}
↓
double f(double x) {
        double r1141021 = 1.0;
        double r1141022 = x;
        double r1141023 = r1141021 / r1141022;
        double r1141024 = r1141022 * r1141022;
        double r1141025 = r1141021 - r1141024;
        double r1141026 = sqrt(r1141025);
        double r1141027 = r1141026 / r1141022;
        double r1141028 = r1141023 + r1141027;
        double r1141029 = sqrt(r1141028);
        double r1141030 = r1141029 * r1141029;
        double r1141031 = log(r1141030);
        return r1141031;
}

Error

Bits error versus x

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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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))