Average Error: 0.0 → 0.0
Time: 21.2s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\left(\sqrt{1 - x \cdot x} + 1\right) \cdot \frac{1}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\left(\sqrt{1 - x \cdot x} + 1\right) \cdot \frac{1}{x}\right)
double f(double x) {
        double r8491032 = 1.0;
        double r8491033 = x;
        double r8491034 = r8491032 / r8491033;
        double r8491035 = r8491033 * r8491033;
        double r8491036 = r8491032 - r8491035;
        double r8491037 = sqrt(r8491036);
        double r8491038 = r8491037 / r8491033;
        double r8491039 = r8491034 + r8491038;
        double r8491040 = log(r8491039);
        return r8491040;
}
↓
double f(double x) {
        double r8491041 = 1.0;
        double r8491042 = x;
        double r8491043 = r8491042 * r8491042;
        double r8491044 = r8491041 - r8491043;
        double r8491045 = sqrt(r8491044);
        double r8491046 = r8491045 + r8491041;
        double r8491047 = r8491041 / r8491042;
        double r8491048 = r8491046 * r8491047;
        double r8491049 = log(r8491048);
        return r8491049;
}

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 div-inv0.0

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied times-frac0.0

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

  7. Applied distribute-rgt-out0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))