Average Error: 0.0 → 0.0
Time: 2.8s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{\mathsf{fma}\left(2 - \frac{2}{1 \cdot \left(1 + t\right)}, 2 - \frac{2}{1 \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{1 \cdot \left(1 + t\right)}, 2 - \frac{2}{1 \cdot \left(1 + t\right)}, 2\right)}\]
\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}{1 \cdot \left(1 + t\right)}, 2 - \frac{2}{1 \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{1 \cdot \left(1 + t\right)}, 2 - \frac{2}{1 \cdot \left(1 + t\right)}, 2\right)}
double f(double t) {
        double r34527 = 1.0;
        double r34528 = 2.0;
        double r34529 = t;
        double r34530 = r34528 / r34529;
        double r34531 = r34527 / r34529;
        double r34532 = r34527 + r34531;
        double r34533 = r34530 / r34532;
        double r34534 = r34528 - r34533;
        double r34535 = r34534 * r34534;
        double r34536 = r34527 + r34535;
        double r34537 = r34528 + r34535;
        double r34538 = r34536 / r34537;
        return r34538;
}
↓
double f(double t) {
        double r34539 = 2.0;
        double r34540 = 1.0;
        double r34541 = 1.0;
        double r34542 = t;
        double r34543 = r34541 + r34542;
        double r34544 = r34540 * r34543;
        double r34545 = r34539 / r34544;
        double r34546 = r34539 - r34545;
        double r34547 = fma(r34546, r34546, r34540);
        double r34548 = fma(r34546, r34546, r34539);
        double r34549 = r34547 / r34548;
        return r34549;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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))))))))