Average Error: 0.0 → 0.0
Time: 4.0s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1} + x}}{\sqrt{x}} \cdot \frac{\sqrt{\sqrt{1} - x}}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1} + x}}{\sqrt{x}} \cdot \frac{\sqrt{\sqrt{1} - x}}{\sqrt{x}}\right)
double f(double x) {
        double r92218 = 1.0;
        double r92219 = x;
        double r92220 = r92218 / r92219;
        double r92221 = r92219 * r92219;
        double r92222 = r92218 - r92221;
        double r92223 = sqrt(r92222);
        double r92224 = r92223 / r92219;
        double r92225 = r92220 + r92224;
        double r92226 = log(r92225);
        return r92226;
}
↓
double f(double x) {
        double r92227 = 1.0;
        double r92228 = x;
        double r92229 = r92227 / r92228;
        double r92230 = sqrt(r92227);
        double r92231 = r92230 + r92228;
        double r92232 = sqrt(r92231);
        double r92233 = sqrt(r92228);
        double r92234 = r92232 / r92233;
        double r92235 = r92230 - r92228;
        double r92236 = sqrt(r92235);
        double r92237 = r92236 / r92233;
        double r92238 = r92234 * r92237;
        double r92239 = r92229 + r92238;
        double r92240 = log(r92239);
        return r92240;
}

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 add-sqr-sqrt0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied difference-of-squares0.0

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

  6. Applied sqrt-prod0.0

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

  7. Applied times-frac0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))