Average Error: 0.1 → 0.1
Time: 11.8s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\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}}\right) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)
double f(double x) {
        double r2394124 = 1.0;
        double r2394125 = x;
        double r2394126 = r2394124 / r2394125;
        double r2394127 = r2394125 * r2394125;
        double r2394128 = r2394124 - r2394127;
        double r2394129 = sqrt(r2394128);
        double r2394130 = r2394129 / r2394125;
        double r2394131 = r2394126 + r2394130;
        double r2394132 = log(r2394131);
        return r2394132;
}
↓
double f(double x) {
        double r2394133 = 1.0;
        double r2394134 = x;
        double r2394135 = r2394133 / r2394134;
        double r2394136 = r2394134 * r2394134;
        double r2394137 = r2394133 - r2394136;
        double r2394138 = sqrt(r2394137);
        double r2394139 = r2394138 / r2394134;
        double r2394140 = r2394135 + r2394139;
        double r2394141 = sqrt(r2394140);
        double r2394142 = log(r2394141);
        double r2394143 = r2394142 + r2394142;
        return r2394143;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.1

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

  4. Applied log-prod0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))