Average Error: 0.0 → 0.0
Time: 3.2s
Precision: binary64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
↓
\[\frac{\mathsf{fma}\left(t, \frac{4}{2 + \left(t + \frac{1}{t}\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{4}{\frac{1}{t} + \left(t + 2\right)}, 2\right)} \]
\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(t, \frac{4}{2 + \left(t + \frac{1}{t}\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{4}{\frac{1}{t} + \left(t + 2\right)}, 2\right)}

(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
 (/
  (fma t (/ 4.0 (+ 2.0 (+ t (/ 1.0 t)))) 1.0)
  (fma t (/ 4.0 (+ (/ 1.0 t) (+ t 2.0))) 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) {
	return fma(t, (4.0 / (2.0 + (t + (1.0 / t)))), 1.0) / fma(t, (4.0 / ((1.0 / t) + (t + 2.0))), 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. Applied associate-+r+_binary640.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022125 
(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))))))