Average Error: 0.0 → 0.0
Time: 2.3s
Precision: binary64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
↓
\[\begin{array}{l} t_1 := \frac{4}{2 + \left(t + \frac{1}{t}\right)}\\ \frac{\mathsf{fma}\left(t, t_1, 1\right)}{\mathsf{fma}\left(t, t_1, 2\right)} \end{array} \]
\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}}
↓
\begin{array}{l}
t_1 := \frac{4}{2 + \left(t + \frac{1}{t}\right)}\\
\frac{\mathsf{fma}\left(t, t_1, 1\right)}{\mathsf{fma}\left(t, t_1, 2\right)}
\end{array}

(FPCore (t)
 :precision binary64
 (/
  (+ 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))))))
↓
(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 4.0 (+ 2.0 (+ t (/ 1.0 t))))))
   (/ (fma t t_1 1.0) (fma t t_1 2.0))))
double code(double t) {
	return (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))));
}
↓
double code(double t) {
	double t_1 = 4.0 / (2.0 + (t + (1.0 / t)));
	return fma(t, t_1, 1.0) / fma(t, t_1, 2.0);
}

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

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, \frac{4}{2 + \left(t + \frac{1}{t}\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{4}{2 + \left(t + \frac{1}{t}\right)}, 2\right)}} \]

  3. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(t, \frac{4}{2 + \left(t + \frac{1}{t}\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{4}{2 + \left(t + \frac{1}{t}\right)}, 2\right)} \]

Reproduce

herbie shell --seed 2021275 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 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))))))