Average Error: 0.0 → 0.0

Time: 5.3s

Precision: binary64

Cost: 1728

\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

↓

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

\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}{2 + \left(t + \frac{1}{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 (+ 2.0 (+ t (/ 1.0 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 / (2.0 + (t + (1.0 / t)))))) / (2.0 + (t * (4.0 / (2.0 + (t + (1.0 / t))))));
}

Error

The X axis uses an exponential scale — Bits error versus `t`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	1.3
Cost	1025

\[\begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 1.712769394525827 \cdot 10^{-14}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array}\]

Alternative 2
Error	1.0
Cost	706

\[\begin{array}{l} \mathbf{if}\;t \leq -0.3261654012971321:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1.000165277867602:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array}\]

Alternative 3
Error	26.2
Cost	64

\[0.8333333333333334\]

Alternative 4
Error	51.2
Cost	64

\[1\]

Derivation

Initial program 0.0
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
Simplified0.0
\[\leadsto \color{blue}{\frac{1 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}{2 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}}\]
Simplified0.0
\[\leadsto \color{blue}{\frac{1 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}{2 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}}\]
Final simplification0.0
\[\leadsto \frac{1 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}{2 + t \cdot \frac{4}{2 + \left(t + \frac{1}{t}\right)}}\]

Reproduce

herbie shell --seed 2021044 
(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))))))