Hyperbolic arc-(co)secant

Average Error: 0.1 → 0.1

Time: 17.9s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r3000129 = 1.0;
        double r3000130 = x;
        double r3000131 = r3000129 / r3000130;
        double r3000132 = r3000130 * r3000130;
        double r3000133 = r3000129 - r3000132;
        double r3000134 = sqrt(r3000133);
        double r3000135 = r3000134 / r3000130;
        double r3000136 = r3000131 + r3000135;
        double r3000137 = log(r3000136);
        return r3000137;
}

↓

double f(double x) {
        double r3000138 = 1.0;
        double r3000139 = x;
        double r3000140 = r3000139 * r3000139;
        double r3000141 = r3000138 - r3000140;
        double r3000142 = sqrt(r3000141);
        double r3000143 = r3000138 + r3000142;
        double r3000144 = 1.0;
        double r3000145 = r3000144 / r3000139;
        double r3000146 = r3000143 * r3000145;
        double r3000147 = log(r3000146);
        return r3000147;
}

Error

Bits error versus x

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-inv 0.1

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

4. Applied div-inv 0.1

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

5. Applied distribute-rgt-out 0.1

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

6. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))