Kahan p13 Example 2

Average Error: 0.0 → 0.0

Time: 17.3s

Precision: 64

Math

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

↓

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

C

double f(double t) {
        double r102428 = 1.0;
        double r102429 = 2.0;
        double r102430 = t;
        double r102431 = r102429 / r102430;
        double r102432 = r102428 / r102430;
        double r102433 = r102428 + r102432;
        double r102434 = r102431 / r102433;
        double r102435 = r102429 - r102434;
        double r102436 = r102435 * r102435;
        double r102437 = r102428 + r102436;
        double r102438 = r102429 + r102436;
        double r102439 = r102437 / r102438;
        return r102439;
}

↓

double f(double t) {
        double r102440 = 1.0;
        double r102441 = 2.0;
        double r102442 = t;
        double r102443 = r102441 / r102442;
        double r102444 = cbrt(r102443);
        double r102445 = r102440 / r102442;
        double r102446 = r102440 + r102445;
        double r102447 = cbrt(r102446);
        double r102448 = r102444 / r102447;
        double r102449 = 3.0;
        double r102450 = pow(r102448, r102449);
        double r102451 = r102441 - r102450;
        double r102452 = r102444 * r102444;
        double r102453 = r102447 * r102447;
        double r102454 = r102452 / r102453;
        double r102455 = -r102448;
        double r102456 = r102455 + r102448;
        double r102457 = r102454 * r102456;
        double r102458 = r102451 + r102457;
        double r102459 = r102443 / r102446;
        double r102460 = r102441 - r102459;
        double r102461 = r102458 * r102460;
        double r102462 = r102440 + r102461;
        double r102463 = 1.0;
        double r102464 = -r102450;
        double r102465 = fma(r102463, r102441, r102464);
        double r102466 = fma(r102464, r102463, r102450);
        double r102467 = r102465 + r102466;
        double r102468 = r102460 * r102467;
        double r102469 = r102441 + r102468;
        double r102470 = r102462 / r102469;
        return r102470;
}

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. Using strategy rm

3. Applied add-cube-cbrt 0.0

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

4. Applied add-cube-cbrt 0.0

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

5. Applied times-frac 0.0

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

6. Applied add-sqr-sqrt 0.5

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

7. Applied prod-diff 0.5

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

8. Simplified 0.0

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

9. Simplified 0.0

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

11. Applied add-cube-cbrt 0.0

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

12. Applied add-cube-cbrt 0.0

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

13. Applied times-frac 0.0

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

14. Applied *-un-lft-identity 0.0

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

15. Applied prod-diff 0.0

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

16. Simplified 0.0

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

17. Simplified 0.0

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

18. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020056 +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))))))))