Average Error: 0.0 → 0.0
Time: 7.3s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 1\right)}{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 2\right)}\]
\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}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 1\right)}{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 2\right)}
double f(double t) {
        double r1525577 = 1.0;
        double r1525578 = 2.0;
        double r1525579 = t;
        double r1525580 = r1525578 * r1525579;
        double r1525581 = r1525577 + r1525579;
        double r1525582 = r1525580 / r1525581;
        double r1525583 = r1525582 * r1525582;
        double r1525584 = r1525577 + r1525583;
        double r1525585 = r1525578 + r1525583;
        double r1525586 = r1525584 / r1525585;
        return r1525586;
}
↓
double f(double t) {
        double r1525587 = 2.0;
        double r1525588 = 1.0;
        double r1525589 = t;
        double r1525590 = r1525588 + r1525589;
        double r1525591 = r1525590 / r1525589;
        double r1525592 = r1525587 / r1525591;
        double r1525593 = fma(r1525592, r1525592, r1525588);
        double r1525594 = fma(r1525592, r1525592, r1525587);
        double r1525595 = r1525593 / r1525594;
        return r1525595;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 +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))))))