Average Error: 0.1 → 0.1
Time: 9.0s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\mathsf{fma}\left(\left(\frac{t \cdot 2}{1 + t}\right), \left(\frac{t \cdot 2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(\frac{t \cdot 2}{1 + t}\right), \left(\frac{t \cdot 2}{1 + t}\right), 2\right)}\]
\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(\left(\frac{t \cdot 2}{1 + t}\right), \left(\frac{t \cdot 2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(\frac{t \cdot 2}{1 + t}\right), \left(\frac{t \cdot 2}{1 + t}\right), 2\right)}
double f(double t) {
        double r3907187 = 1.0;
        double r3907188 = 2.0;
        double r3907189 = t;
        double r3907190 = r3907188 * r3907189;
        double r3907191 = r3907187 + r3907189;
        double r3907192 = r3907190 / r3907191;
        double r3907193 = r3907192 * r3907192;
        double r3907194 = r3907187 + r3907193;
        double r3907195 = r3907188 + r3907193;
        double r3907196 = r3907194 / r3907195;
        return r3907196;
}
↓
double f(double t) {
        double r3907197 = t;
        double r3907198 = 2.0;
        double r3907199 = r3907197 * r3907198;
        double r3907200 = 1.0;
        double r3907201 = r3907200 + r3907197;
        double r3907202 = r3907199 / r3907201;
        double r3907203 = fma(r3907202, r3907202, r3907200);
        double r3907204 = fma(r3907202, r3907202, r3907198);
        double r3907205 = r3907203 / r3907204;
        return r3907205;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.1

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019121 +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))))))