Average Error: 0.0 → 0.0
Time: 16.6s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\mathsf{fma}\left(\left(\frac{t \cdot 2}{1 + t}\right), \left(\frac{t \cdot 2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(\frac{t \cdot 2}{1 + t}\right), \left(\frac{t \cdot 2}{1 + t}\right), 2\right)}\]
\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(\left(\frac{t \cdot 2}{1 + t}\right), \left(\frac{t \cdot 2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(\frac{t \cdot 2}{1 + t}\right), \left(\frac{t \cdot 2}{1 + t}\right), 2\right)}
double f(double t) {
        double r4030018 = 1.0;
        double r4030019 = 2.0;
        double r4030020 = t;
        double r4030021 = r4030019 * r4030020;
        double r4030022 = r4030018 + r4030020;
        double r4030023 = r4030021 / r4030022;
        double r4030024 = r4030023 * r4030023;
        double r4030025 = r4030018 + r4030024;
        double r4030026 = r4030019 + r4030024;
        double r4030027 = r4030025 / r4030026;
        return r4030027;
}
↓
double f(double t) {
        double r4030028 = t;
        double r4030029 = 2.0;
        double r4030030 = r4030028 * r4030029;
        double r4030031 = 1.0;
        double r4030032 = r4030031 + r4030028;
        double r4030033 = r4030030 / r4030032;
        double r4030034 = fma(r4030033, r4030033, r4030031);
        double r4030035 = fma(r4030033, r4030033, r4030029);
        double r4030036 = r4030034 / r4030035;
        return r4030036;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019125 +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))))))