Average Error: 0.0 → 0.0
Time: 11.2s
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 r2566962 = 1.0;
        double r2566963 = 2.0;
        double r2566964 = t;
        double r2566965 = r2566963 * r2566964;
        double r2566966 = r2566962 + r2566964;
        double r2566967 = r2566965 / r2566966;
        double r2566968 = r2566967 * r2566967;
        double r2566969 = r2566962 + r2566968;
        double r2566970 = r2566963 + r2566968;
        double r2566971 = r2566969 / r2566970;
        return r2566971;
}
↓
double f(double t) {
        double r2566972 = 2.0;
        double r2566973 = t;
        double r2566974 = r2566972 * r2566973;
        double r2566975 = 1.0;
        double r2566976 = r2566975 + r2566973;
        double r2566977 = r2566974 / r2566976;
        double r2566978 = fma(r2566977, r2566977, r2566975);
        double r2566979 = fma(r2566977, r2566977, r2566972);
        double r2566980 = r2566978 / r2566979;
        return r2566980;
}

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

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

  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 2019200 +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))))))