Average Error: 0.0 → 0.0
Time: 1.7s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, \frac{2 \cdot t}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{2}{1 + t}, t \cdot \frac{t}{1 + t}, 1\right)}}{2}\]
\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{\frac{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, \frac{2 \cdot t}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{2}{1 + t}, t \cdot \frac{t}{1 + t}, 1\right)}}{2}
double f(double t) {
        double r57603 = 1.0;
        double r57604 = 2.0;
        double r57605 = t;
        double r57606 = r57604 * r57605;
        double r57607 = r57603 + r57605;
        double r57608 = r57606 / r57607;
        double r57609 = r57608 * r57608;
        double r57610 = r57603 + r57609;
        double r57611 = r57604 + r57609;
        double r57612 = r57610 / r57611;
        return r57612;
}
↓
double f(double t) {
        double r57613 = 2.0;
        double r57614 = t;
        double r57615 = r57613 * r57614;
        double r57616 = 1.0;
        double r57617 = r57616 + r57614;
        double r57618 = r57615 / r57617;
        double r57619 = fma(r57618, r57618, r57616);
        double r57620 = r57613 / r57617;
        double r57621 = r57614 / r57617;
        double r57622 = r57614 * r57621;
        double r57623 = 1.0;
        double r57624 = fma(r57620, r57622, r57623);
        double r57625 = r57619 / r57624;
        double r57626 = r57625 / r57613;
        return r57626;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))