Average Error: 0.0 → 0.1
Time: 1.1s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, \frac{2 \cdot t}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{2}{1 + t}, t \cdot \frac{t}{1 + t}, 1\right)}}{2}\]
\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{\frac{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, \frac{2 \cdot t}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{2}{1 + t}, t \cdot \frac{t}{1 + t}, 1\right)}}{2}
double f(double t) {
        double r54747 = 1.0;
        double r54748 = 2.0;
        double r54749 = t;
        double r54750 = r54748 * r54749;
        double r54751 = r54747 + r54749;
        double r54752 = r54750 / r54751;
        double r54753 = r54752 * r54752;
        double r54754 = r54747 + r54753;
        double r54755 = r54748 + r54753;
        double r54756 = r54754 / r54755;
        return r54756;
}
↓
double f(double t) {
        double r54757 = 2.0;
        double r54758 = t;
        double r54759 = r54757 * r54758;
        double r54760 = 1.0;
        double r54761 = r54760 + r54758;
        double r54762 = r54759 / r54761;
        double r54763 = fma(r54762, r54762, r54760);
        double r54764 = r54757 / r54761;
        double r54765 = r54758 / r54761;
        double r54766 = r54758 * r54765;
        double r54767 = 1.0;
        double r54768 = fma(r54764, r54766, r54767);
        double r54769 = r54763 / r54768;
        double r54770 = r54769 / r54757;
        return r54770;
}

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. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020003 +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))))))