Average Error: 0.0 → 0.0
Time: 4.2s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, \frac{2 \cdot t}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{2}{1 + t}, t \cdot \frac{t}{1 + t}, 1\right)}}{2}\]
\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 code(double t) {
	return ((double) (((double) (1.0 + ((double) (((double) (((double) (2.0 * t)) / ((double) (1.0 + t)))) * ((double) (((double) (2.0 * t)) / ((double) (1.0 + t)))))))) / ((double) (2.0 + ((double) (((double) (((double) (2.0 * t)) / ((double) (1.0 + t)))) * ((double) (((double) (2.0 * t)) / ((double) (1.0 + t))))))))));
}
↓
double code(double t) {
	return ((double) (((double) (((double) fma(((double) (((double) (2.0 * t)) / ((double) (1.0 + t)))), ((double) (((double) (2.0 * t)) / ((double) (1.0 + t)))), 1.0)) / ((double) fma(((double) (2.0 / ((double) (1.0 + t)))), ((double) (t * ((double) (t / ((double) (1.0 + t)))))), 1.0)))) / 2.0));
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020120 +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))))))