Average Error: 0.0 → 0.0
Time: 15.1s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\mathsf{fma}\left(\frac{2 \cdot t}{t + 1}, \frac{2 \cdot t}{t + 1}, 1\right)}{\mathsf{fma}\left(\frac{2 \cdot t}{t + 1}, \frac{2 \cdot t}{t + 1}, 2\right)}\]
\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{2 \cdot t}{t + 1}, \frac{2 \cdot t}{t + 1}, 1\right)}{\mathsf{fma}\left(\frac{2 \cdot t}{t + 1}, \frac{2 \cdot t}{t + 1}, 2\right)}
double f(double t) {
        double r2349173 = 1.0;
        double r2349174 = 2.0;
        double r2349175 = t;
        double r2349176 = r2349174 * r2349175;
        double r2349177 = r2349173 + r2349175;
        double r2349178 = r2349176 / r2349177;
        double r2349179 = r2349178 * r2349178;
        double r2349180 = r2349173 + r2349179;
        double r2349181 = r2349174 + r2349179;
        double r2349182 = r2349180 / r2349181;
        return r2349182;
}
↓
double f(double t) {
        double r2349183 = 2.0;
        double r2349184 = t;
        double r2349185 = r2349183 * r2349184;
        double r2349186 = 1.0;
        double r2349187 = r2349184 + r2349186;
        double r2349188 = r2349185 / r2349187;
        double r2349189 = fma(r2349188, r2349188, r2349186);
        double r2349190 = fma(r2349188, r2349188, r2349183);
        double r2349191 = r2349189 / r2349190;
        return r2349191;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 1"
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))