Average Error: 0.0 → 0.0
Time: 16.5s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{\mathsf{fma}\left(2 - \frac{2}{t + 1}, \mathsf{fma}\left(\frac{-1}{\sqrt[3]{t + 1} \cdot \sqrt[3]{t + 1}}, \frac{2}{\sqrt[3]{t + 1}}, \frac{\frac{2}{\sqrt[3]{t + 1}}}{\sqrt[3]{t + 1} \cdot \sqrt[3]{t + 1}}\right) + \left(2 - \frac{\frac{2}{\sqrt[3]{t + 1}}}{\sqrt[3]{t + 1} \cdot \sqrt[3]{t + 1}}\right), 1\right)}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 2\right)}\]
\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}{t + 1}, \mathsf{fma}\left(\frac{-1}{\sqrt[3]{t + 1} \cdot \sqrt[3]{t + 1}}, \frac{2}{\sqrt[3]{t + 1}}, \frac{\frac{2}{\sqrt[3]{t + 1}}}{\sqrt[3]{t + 1} \cdot \sqrt[3]{t + 1}}\right) + \left(2 - \frac{\frac{2}{\sqrt[3]{t + 1}}}{\sqrt[3]{t + 1} \cdot \sqrt[3]{t + 1}}\right), 1\right)}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 2\right)}
double f(double t) {
        double r879246 = 1.0;
        double r879247 = 2.0;
        double r879248 = t;
        double r879249 = r879247 / r879248;
        double r879250 = r879246 / r879248;
        double r879251 = r879246 + r879250;
        double r879252 = r879249 / r879251;
        double r879253 = r879247 - r879252;
        double r879254 = r879253 * r879253;
        double r879255 = r879246 + r879254;
        double r879256 = r879247 + r879254;
        double r879257 = r879255 / r879256;
        return r879257;
}
↓
double f(double t) {
        double r879258 = 2.0;
        double r879259 = t;
        double r879260 = 1.0;
        double r879261 = r879259 + r879260;
        double r879262 = r879258 / r879261;
        double r879263 = r879258 - r879262;
        double r879264 = -1.0;
        double r879265 = cbrt(r879261);
        double r879266 = r879265 * r879265;
        double r879267 = r879264 / r879266;
        double r879268 = r879258 / r879265;
        double r879269 = r879268 / r879266;
        double r879270 = fma(r879267, r879268, r879269);
        double r879271 = r879258 - r879269;
        double r879272 = r879270 + r879271;
        double r879273 = fma(r879263, r879272, r879260);
        double r879274 = fma(r879263, r879263, r879258);
        double r879275 = r879273 / r879274;
        return r879275;
}

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

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied times-frac0.0

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

  7. Applied add-cube-cbrt0.8

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

  8. Applied prod-diff0.8

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

  9. Simplified0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 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))))))))