Average Error: 0.0 → 0.0
Time: 27.7s
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 r2174045 = 1.0;
        double r2174046 = 2.0;
        double r2174047 = t;
        double r2174048 = r2174046 * r2174047;
        double r2174049 = r2174045 + r2174047;
        double r2174050 = r2174048 / r2174049;
        double r2174051 = r2174050 * r2174050;
        double r2174052 = r2174045 + r2174051;
        double r2174053 = r2174046 + r2174051;
        double r2174054 = r2174052 / r2174053;
        return r2174054;
}
↓
double f(double t) {
        double r2174055 = t;
        double r2174056 = 2.0;
        double r2174057 = r2174055 * r2174056;
        double r2174058 = 1.0;
        double r2174059 = r2174058 + r2174055;
        double r2174060 = r2174057 / r2174059;
        double r2174061 = fma(r2174060, r2174060, r2174058);
        double r2174062 = fma(r2174060, r2174060, r2174056);
        double r2174063 = r2174061 / r2174062;
        return r2174063;
}

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 2019134 +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))))))