Average Error: 0.1 → 0.1
Time: 10.7s
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 r1539687 = 1.0;
        double r1539688 = x;
        double r1539689 = r1539687 / r1539688;
        double r1539690 = r1539688 * r1539688;
        double r1539691 = r1539687 - r1539690;
        double r1539692 = sqrt(r1539691);
        double r1539693 = r1539692 / r1539688;
        double r1539694 = r1539689 + r1539693;
        double r1539695 = log(r1539694);
        return r1539695;
}
↓
double f(double x) {
        double r1539696 = 1.0;
        double r1539697 = x;
        double r1539698 = r1539697 * r1539697;
        double r1539699 = r1539696 - r1539698;
        double r1539700 = sqrt(r1539699);
        double r1539701 = r1539700 + r1539696;
        double r1539702 = r1539696 / r1539697;
        double r1539703 = r1539701 * r1539702;
        double r1539704 = log(r1539703);
        return r1539704;
}

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 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))