Average Error: 0.0 → 0.0
Time: 4.8s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \sqrt[3]{{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}}\]
1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \sqrt[3]{{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}}
double f(double t) {
        double r49159 = 1.0;
        double r49160 = 2.0;
        double r49161 = t;
        double r49162 = r49160 / r49161;
        double r49163 = r49159 / r49161;
        double r49164 = r49159 + r49163;
        double r49165 = r49162 / r49164;
        double r49166 = r49160 - r49165;
        double r49167 = r49166 * r49166;
        double r49168 = r49160 + r49167;
        double r49169 = r49159 / r49168;
        double r49170 = r49159 - r49169;
        return r49170;
}
↓
double f(double t) {
        double r49171 = 1.0;
        double r49172 = 2.0;
        double r49173 = t;
        double r49174 = r49172 / r49173;
        double r49175 = r49171 / r49173;
        double r49176 = r49171 + r49175;
        double r49177 = r49174 / r49176;
        double r49178 = r49172 - r49177;
        double r49179 = 3.0;
        double r49180 = pow(r49178, r49179);
        double r49181 = cbrt(r49180);
        double r49182 = r49178 * r49181;
        double r49183 = r49172 + r49182;
        double r49184 = r49171 / r49183;
        double r49185 = r49171 - r49184;
        return r49185;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube0.0

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\sqrt[3]{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}\]

  4. Simplified0.0

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \sqrt[3]{\color{blue}{{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}}}\]

  5. Final simplification0.0

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \sqrt[3]{{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))