Average Error: 0.1 → 0.1
Time: 11.7s
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 r1527189 = 1.0;
        double r1527190 = x;
        double r1527191 = r1527189 / r1527190;
        double r1527192 = r1527190 * r1527190;
        double r1527193 = r1527189 - r1527192;
        double r1527194 = sqrt(r1527193);
        double r1527195 = r1527194 / r1527190;
        double r1527196 = r1527191 + r1527195;
        double r1527197 = log(r1527196);
        return r1527197;
}
↓
double f(double x) {
        double r1527198 = 1.0;
        double r1527199 = x;
        double r1527200 = r1527199 * r1527199;
        double r1527201 = r1527198 - r1527200;
        double r1527202 = sqrt(r1527201);
        double r1527203 = r1527202 + r1527198;
        double r1527204 = r1527198 / r1527199;
        double r1527205 = r1527203 * r1527204;
        double r1527206 = log(r1527205);
        return r1527206;
}

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 div-inv0.1

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

  4. Applied *-un-lft-identity0.1

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

  5. Applied *-un-lft-identity0.1

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

  6. Applied times-frac0.1

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

  7. Applied distribute-rgt-out0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))