Average Error: 0.1 → 0.2
Time: 14.9s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\left(-\log \left(\sqrt{x}\right)\right) + \log \left(\frac{1}{\sqrt{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\left(-\log \left(\sqrt{x}\right)\right) + \log \left(\frac{1}{\sqrt{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{x}}\right)
double f(double x) {
        double r72371 = 1.0;
        double r72372 = x;
        double r72373 = r72371 / r72372;
        double r72374 = r72372 * r72372;
        double r72375 = r72371 - r72374;
        double r72376 = sqrt(r72375);
        double r72377 = r72376 / r72372;
        double r72378 = r72373 + r72377;
        double r72379 = log(r72378);
        return r72379;
}
↓
double f(double x) {
        double r72380 = x;
        double r72381 = sqrt(r72380);
        double r72382 = log(r72381);
        double r72383 = -r72382;
        double r72384 = 1.0;
        double r72385 = r72384 / r72381;
        double r72386 = r72380 * r72380;
        double r72387 = r72384 - r72386;
        double r72388 = sqrt(r72387);
        double r72389 = r72388 / r72381;
        double r72390 = r72385 + r72389;
        double r72391 = log(r72390);
        double r72392 = r72383 + r72391;
        return r72392;
}

Error

Bits error versus x

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Results

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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

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

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

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

  5. Applied times-frac0.1

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

  6. Applied add-sqr-sqrt0.1

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

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

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

  8. Applied times-frac0.1

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

  9. Applied distribute-lft-out0.1

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

  10. Applied log-prod0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))