Average Error: 0.0 → 0.0
Time: 5.3s
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 r1692813 = 1.0;
        double r1692814 = 2.0;
        double r1692815 = t;
        double r1692816 = r1692814 * r1692815;
        double r1692817 = r1692813 + r1692815;
        double r1692818 = r1692816 / r1692817;
        double r1692819 = r1692818 * r1692818;
        double r1692820 = r1692813 + r1692819;
        double r1692821 = r1692814 + r1692819;
        double r1692822 = r1692820 / r1692821;
        return r1692822;
}
↓
double f(double t) {
        double r1692823 = 2.0;
        double r1692824 = t;
        double r1692825 = r1692823 * r1692824;
        double r1692826 = 1.0;
        double r1692827 = r1692824 + r1692826;
        double r1692828 = r1692825 / r1692827;
        double r1692829 = fma(r1692828, r1692828, r1692826);
        double r1692830 = fma(r1692828, r1692828, r1692823);
        double r1692831 = r1692829 / r1692830;
        return r1692831;
}

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 2019171 +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))))))