Average Error: 0.1 → 0.1
Time: 12.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1 - x \cdot x}}}{\sqrt{x}} \cdot \frac{\sqrt{\sqrt{1 - x \cdot x}}}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1 - x \cdot x}}}{\sqrt{x}} \cdot \frac{\sqrt{\sqrt{1 - x \cdot x}}}{\sqrt{x}}\right)
double f(double x) {
        double r58948 = 1.0;
        double r58949 = x;
        double r58950 = r58948 / r58949;
        double r58951 = r58949 * r58949;
        double r58952 = r58948 - r58951;
        double r58953 = sqrt(r58952);
        double r58954 = r58953 / r58949;
        double r58955 = r58950 + r58954;
        double r58956 = log(r58955);
        return r58956;
}
↓
double f(double x) {
        double r58957 = 1.0;
        double r58958 = x;
        double r58959 = r58957 / r58958;
        double r58960 = r58958 * r58958;
        double r58961 = r58957 - r58960;
        double r58962 = sqrt(r58961);
        double r58963 = sqrt(r58962);
        double r58964 = sqrt(r58958);
        double r58965 = r58963 / r58964;
        double r58966 = r58965 * r58965;
        double r58967 = r58959 + r58966;
        double r58968 = log(r58967);
        return r58968;
}

Error

Bits error versus x

Try it out

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 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 add-sqr-sqrt0.1

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

  5. Applied sqrt-prod0.1

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

  6. Applied times-frac0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))