Average Error: 0.0 → 0.0
Time: 16.2s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, \frac{2 \cdot t}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, \frac{2 \cdot t}{1 + t}, 2\right)}\]
\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}{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)}
double f(double t) {
        double r53070 = 1.0;
        double r53071 = 2.0;
        double r53072 = t;
        double r53073 = r53071 * r53072;
        double r53074 = r53070 + r53072;
        double r53075 = r53073 / r53074;
        double r53076 = r53075 * r53075;
        double r53077 = r53070 + r53076;
        double r53078 = r53071 + r53076;
        double r53079 = r53077 / r53078;
        return r53079;
}
↓
double f(double t) {
        double r53080 = 2.0;
        double r53081 = t;
        double r53082 = r53080 * r53081;
        double r53083 = 1.0;
        double r53084 = r53083 + r53081;
        double r53085 = r53082 / r53084;
        double r53086 = fma(r53085, r53085, r53083);
        double r53087 = fma(r53085, r53085, r53080);
        double r53088 = r53086 / r53087;
        return r53088;
}

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

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

Reproduce

herbie shell --seed 2019323 +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))))))