Average Error: 0.0 → 0.0
Time: 15.7s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \left(\left(2 + \left(-{\left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)}^{3}\right)\right) + \left(\left(-{\left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)}^{3}\right) + {\left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)}^{3}\right)\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\frac{1 + \left(\left(2 + \left(-{\left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)}^{3}\right)\right) + \left(\left(-{\left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)}^{3}\right) + {\left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)}^{3}\right)\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
double f(double t) {
        double r60128 = 1.0;
        double r60129 = 2.0;
        double r60130 = t;
        double r60131 = r60129 / r60130;
        double r60132 = r60128 / r60130;
        double r60133 = r60128 + r60132;
        double r60134 = r60131 / r60133;
        double r60135 = r60129 - r60134;
        double r60136 = r60135 * r60135;
        double r60137 = r60128 + r60136;
        double r60138 = r60129 + r60136;
        double r60139 = r60137 / r60138;
        return r60139;
}
↓
double f(double t) {
        double r60140 = 1.0;
        double r60141 = 2.0;
        double r60142 = t;
        double r60143 = r60141 / r60142;
        double r60144 = cbrt(r60143);
        double r60145 = r60140 / r60142;
        double r60146 = r60140 + r60145;
        double r60147 = cbrt(r60146);
        double r60148 = r60144 / r60147;
        double r60149 = 3.0;
        double r60150 = pow(r60148, r60149);
        double r60151 = -r60150;
        double r60152 = r60141 + r60151;
        double r60153 = r60151 + r60150;
        double r60154 = r60152 + r60153;
        double r60155 = r60143 / r60146;
        double r60156 = r60141 - r60155;
        double r60157 = r60154 * r60156;
        double r60158 = r60140 + r60157;
        double r60159 = r60156 * r60156;
        double r60160 = r60141 + r60159;
        double r60161 = r60158 / r60160;
        return r60161;
}

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

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

  3. Applied add-cube-cbrt0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied times-frac0.0

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied prod-diff0.5

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

  8. Simplified0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))