Average Error: 0.0 → 0.0
Time: 10.2s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, \left(\left(-\frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \left(2 - \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right), 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}\]
\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{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, \left(\left(-\frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \left(2 - \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right), 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}
double f(double t) {
        double r28266 = 1.0;
        double r28267 = 2.0;
        double r28268 = t;
        double r28269 = r28267 / r28268;
        double r28270 = r28266 / r28268;
        double r28271 = r28266 + r28270;
        double r28272 = r28269 / r28271;
        double r28273 = r28267 - r28272;
        double r28274 = r28273 * r28273;
        double r28275 = r28266 + r28274;
        double r28276 = r28267 + r28274;
        double r28277 = r28275 / r28276;
        return r28277;
}
↓
double f(double t) {
        double r28278 = 2.0;
        double r28279 = 1.0;
        double r28280 = t;
        double r28281 = fma(r28279, r28280, r28279);
        double r28282 = r28278 / r28281;
        double r28283 = r28278 - r28282;
        double r28284 = cbrt(r28281);
        double r28285 = 3.0;
        double r28286 = pow(r28284, r28285);
        double r28287 = r28278 / r28286;
        double r28288 = -r28287;
        double r28289 = r28288 + r28287;
        double r28290 = r28278 - r28287;
        double r28291 = r28289 + r28290;
        double r28292 = fma(r28283, r28291, r28279);
        double r28293 = fma(r28283, r28283, r28278);
        double r28294 = r28292 / r28293;
        return r28294;
}

Error

Bits error versus t

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. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2\right)}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.0

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

  5. Applied add-cube-cbrt0.0

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

  6. Applied times-frac0.0

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

  7. Applied add-cube-cbrt0.8

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

  8. Applied prod-diff0.8

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

  9. Simplified0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))