Average Error: 0.0 → 0.0
Time: 5.4s
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 r1675744 = 1.0;
        double r1675745 = 2.0;
        double r1675746 = t;
        double r1675747 = r1675745 * r1675746;
        double r1675748 = r1675744 + r1675746;
        double r1675749 = r1675747 / r1675748;
        double r1675750 = r1675749 * r1675749;
        double r1675751 = r1675744 + r1675750;
        double r1675752 = r1675745 + r1675750;
        double r1675753 = r1675751 / r1675752;
        return r1675753;
}
↓
double f(double t) {
        double r1675754 = 2.0;
        double r1675755 = t;
        double r1675756 = r1675754 * r1675755;
        double r1675757 = 1.0;
        double r1675758 = r1675755 + r1675757;
        double r1675759 = r1675756 / r1675758;
        double r1675760 = fma(r1675759, r1675759, r1675757);
        double r1675761 = fma(r1675759, r1675759, r1675754);
        double r1675762 = r1675760 / r1675761;
        return r1675762;
}

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 2019174 +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))))))