Average Error: 0.0 → 0.0
Time: 2.4s
Precision: binary64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 + t \cdot \frac{-4}{\frac{-1}{t} + \left(-2 - t\right)}}{2 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}\]
\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{1 + t \cdot \frac{-4}{\frac{-1}{t} + \left(-2 - t\right)}}{2 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\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
 (/
  (+ 1.0 (* t (/ -4.0 (+ (/ -1.0 t) (- -2.0 t)))))
  (+ 2.0 (* t (/ 4.0 (+ 2.0 (+ t (/ 1.0 t))))))))
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 (1.0 + (t * (-4.0 / ((-1.0 / t) + (-2.0 - t))))) / (2.0 + (t * (4.0 / (2.0 + (t + (1.0 / t))))));
}

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

    \[\leadsto \color{blue}{\frac{1 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}{2 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}}\]
  3. Using strategy rm

  4. Applied frac-2neg_binary64_7780.0

    \[\leadsto \frac{1 + t \cdot \color{blue}{\frac{-4}{-\left(2 + \left(t + \frac{1}{t}\right)\right)}}}{2 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}\]

  5. Simplified0.0

    \[\leadsto \frac{1 + t \cdot \frac{\color{blue}{-4}}{-\left(2 + \left(t + \frac{1}{t}\right)\right)}}{2 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}\]

  6. Simplified0.0

    \[\leadsto \frac{1 + t \cdot \frac{-4}{\color{blue}{\frac{-1}{t} + \left(-2 - t\right)}}}{2 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}\]

  7. Final simplification0.0

    \[\leadsto \frac{1 + t \cdot \frac{-4}{\frac{-1}{t} + \left(-2 - t\right)}}{2 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}\]

Reproduce

herbie shell --seed 2020292 
(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))))))