Asymptote B

Average Error: 0.0 → 0.0

Time: 22.9s

Precision: 64

Math

\[\frac{1}{x - 1} + \frac{x}{x + 1}\]

↓

\[\sqrt[3]{\left(\frac{1}{x - 1} + \frac{x}{1 + x}\right) \cdot \frac{\left(\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot \frac{x}{1 + x} + \frac{1}{\left(-1 + x\right) \cdot \left(\left(-1 + x\right) \cdot \left(-1 + x\right)\right)}\right) \cdot \left(\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot \frac{x}{1 + x} + \frac{1}{\left(-1 + x\right) \cdot \left(\left(-1 + x\right) \cdot \left(-1 + x\right)\right)}\right)}{\left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \frac{1}{x - 1} \cdot \frac{x}{1 + x}\right)\right) \cdot \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \frac{1}{x - 1} \cdot \frac{x}{1 + x}\right)\right)}}\]

C

double f(double x) {
        double r4414055 = 1.0;
        double r4414056 = x;
        double r4414057 = r4414056 - r4414055;
        double r4414058 = r4414055 / r4414057;
        double r4414059 = r4414056 + r4414055;
        double r4414060 = r4414056 / r4414059;
        double r4414061 = r4414058 + r4414060;
        return r4414061;
}

↓

double f(double x) {
        double r4414062 = 1.0;
        double r4414063 = x;
        double r4414064 = r4414063 - r4414062;
        double r4414065 = r4414062 / r4414064;
        double r4414066 = r4414062 + r4414063;
        double r4414067 = r4414063 / r4414066;
        double r4414068 = r4414065 + r4414067;
        double r4414069 = r4414067 * r4414067;
        double r4414070 = r4414069 * r4414067;
        double r4414071 = -1.0;
        double r4414072 = r4414071 + r4414063;
        double r4414073 = r4414072 * r4414072;
        double r4414074 = r4414072 * r4414073;
        double r4414075 = r4414062 / r4414074;
        double r4414076 = r4414070 + r4414075;
        double r4414077 = r4414076 * r4414076;
        double r4414078 = r4414065 * r4414065;
        double r4414079 = r4414065 * r4414067;
        double r4414080 = r4414069 - r4414079;
        double r4414081 = r4414078 + r4414080;
        double r4414082 = r4414081 * r4414081;
        double r4414083 = r4414077 / r4414082;
        double r4414084 = r4414068 * r4414083;
        double r4414085 = cbrt(r4414084);
        return r4414085;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

   \[\frac{1}{x - 1} + \frac{x}{x + 1}\]
2. Using strategy rm

3. Applied add-cbrt-cube 0.0

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}}\]
4. Using strategy rm

5. Applied flip3-+ 0.0

   \[\leadsto \sqrt[3]{\left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \color{blue}{\frac{{\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}}{\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{1}{x - 1} \cdot \frac{x}{x + 1}\right)}}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}\]

6. Applied flip3-+ 0.0

   \[\leadsto \sqrt[3]{\left(\color{blue}{\frac{{\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}}{\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{1}{x - 1} \cdot \frac{x}{x + 1}\right)}} \cdot \frac{{\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}}{\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{1}{x - 1} \cdot \frac{x}{x + 1}\right)}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}\]

7. Applied frac-times 0.0

   \[\leadsto \sqrt[3]{\color{blue}{\frac{\left({\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}\right) \cdot \left({\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}\right)}{\left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{1}{x - 1} \cdot \frac{x}{x + 1}\right)\right) \cdot \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{1}{x - 1} \cdot \frac{x}{x + 1}\right)\right)}} \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}\]

8. Simplified 0.0

   \[\leadsto \sqrt[3]{\frac{\color{blue}{\left(\frac{1}{\left(\left(-1 + x\right) \cdot \left(-1 + x\right)\right) \cdot \left(-1 + x\right)} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right)\right) \cdot \left(\frac{1}{\left(\left(-1 + x\right) \cdot \left(-1 + x\right)\right) \cdot \left(-1 + x\right)} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right)\right)}}{\left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{1}{x - 1} \cdot \frac{x}{x + 1}\right)\right) \cdot \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{1}{x - 1} \cdot \frac{x}{x + 1}\right)\right)} \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}\]

9. Final simplification 0.0

   \[\leadsto \sqrt[3]{\left(\frac{1}{x - 1} + \frac{x}{1 + x}\right) \cdot \frac{\left(\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot \frac{x}{1 + x} + \frac{1}{\left(-1 + x\right) \cdot \left(\left(-1 + x\right) \cdot \left(-1 + x\right)\right)}\right) \cdot \left(\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot \frac{x}{1 + x} + \frac{1}{\left(-1 + x\right) \cdot \left(\left(-1 + x\right) \cdot \left(-1 + x\right)\right)}\right)}{\left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \frac{1}{x - 1} \cdot \frac{x}{1 + x}\right)\right) \cdot \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \frac{1}{x - 1} \cdot \frac{x}{1 + x}\right)\right)}}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))