Average Error: 0.0 → 0.0
Time: 7.6s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\mathsf{fma}\left(\frac{t \cdot 2}{1 + t}, \frac{t \cdot 2}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{t \cdot 2}{1 + t}, \frac{t \cdot 2}{1 + t}, 2\right)}\]
\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}
↓
\frac{\mathsf{fma}\left(\frac{t \cdot 2}{1 + t}, \frac{t \cdot 2}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{t \cdot 2}{1 + t}, \frac{t \cdot 2}{1 + t}, 2\right)}
double f(double t) {
        double r1503438 = 1.0;
        double r1503439 = 2.0;
        double r1503440 = t;
        double r1503441 = r1503439 * r1503440;
        double r1503442 = r1503438 + r1503440;
        double r1503443 = r1503441 / r1503442;
        double r1503444 = r1503443 * r1503443;
        double r1503445 = r1503438 + r1503444;
        double r1503446 = r1503439 + r1503444;
        double r1503447 = r1503445 / r1503446;
        return r1503447;
}
↓
double f(double t) {
        double r1503448 = t;
        double r1503449 = 2.0;
        double r1503450 = r1503448 * r1503449;
        double r1503451 = 1.0;
        double r1503452 = r1503451 + r1503448;
        double r1503453 = r1503450 / r1503452;
        double r1503454 = fma(r1503453, r1503453, r1503451);
        double r1503455 = fma(r1503453, r1503453, r1503449);
        double r1503456 = r1503454 / r1503455;
        return r1503456;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 1"
  (/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))