Average Error: 0.0 → 0.0
Time: 7.1s
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 r1801758 = 1.0;
        double r1801759 = 2.0;
        double r1801760 = t;
        double r1801761 = r1801759 * r1801760;
        double r1801762 = r1801758 + r1801760;
        double r1801763 = r1801761 / r1801762;
        double r1801764 = r1801763 * r1801763;
        double r1801765 = r1801758 + r1801764;
        double r1801766 = r1801759 + r1801764;
        double r1801767 = r1801765 / r1801766;
        return r1801767;
}
↓
double f(double t) {
        double r1801768 = 2.0;
        double r1801769 = 1.0;
        double r1801770 = t;
        double r1801771 = r1801769 + r1801770;
        double r1801772 = r1801771 / r1801770;
        double r1801773 = r1801768 / r1801772;
        double r1801774 = fma(r1801773, r1801773, r1801769);
        double r1801775 = fma(r1801773, r1801773, r1801768);
        double r1801776 = r1801774 / r1801775;
        return r1801776;
}

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 2019171 +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))))))