Average Error: 0.0 → 0.0
Time: 15.5s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\mathsf{fma}\left(\frac{t \cdot 2}{1 + t}, \frac{t \cdot 2}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{t \cdot 2}{1 + t}, \frac{t \cdot 2}{1 + t}, 2\right)}\]
\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{t \cdot 2}{1 + t}, \frac{t \cdot 2}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{t \cdot 2}{1 + t}, \frac{t \cdot 2}{1 + t}, 2\right)}
double f(double t) {
        double r1234809 = 1.0;
        double r1234810 = 2.0;
        double r1234811 = t;
        double r1234812 = r1234810 * r1234811;
        double r1234813 = r1234809 + r1234811;
        double r1234814 = r1234812 / r1234813;
        double r1234815 = r1234814 * r1234814;
        double r1234816 = r1234809 + r1234815;
        double r1234817 = r1234810 + r1234815;
        double r1234818 = r1234816 / r1234817;
        return r1234818;
}
↓
double f(double t) {
        double r1234819 = t;
        double r1234820 = 2.0;
        double r1234821 = r1234819 * r1234820;
        double r1234822 = 1.0;
        double r1234823 = r1234822 + r1234819;
        double r1234824 = r1234821 / r1234823;
        double r1234825 = fma(r1234824, r1234824, r1234822);
        double r1234826 = fma(r1234824, r1234824, r1234820);
        double r1234827 = r1234825 / r1234826;
        return r1234827;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019141 +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))))))