Average Error: 0.0 → 0.0

Time: 3.4s

Precision: binary64

Cost: 2240

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

↓

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

\frac{1 + \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 + \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}}

(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 (* (/ (* 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 + (((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 + (((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))));
}

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	16.0
Cost	27392

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

Alternative 2
Error	16.2
Cost	2112

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

Alternative 3
Error	31.9
Cost	1856

\[\frac{1 + \left(t \cdot t\right) \cdot \left(4 - t \cdot 8\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

Alternative 4
Error	0.0
Cost	1728

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

Alternative 5
Error	31.9
Cost	1728

\[\frac{1 + \left(t \cdot t\right) \cdot \left(4 - t \cdot 8\right)}{2 + \frac{4 \cdot \left(t \cdot t\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}}\]

Alternative 6
Error	31.1
Cost	1600

\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \left(4 - \frac{8}{t}\right)}\]

Alternative 7
Error	31.0
Cost	1600

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

Alternative 8
Error	30.0
Cost	1472

\[\frac{1 + \left(t \cdot t\right) \cdot \left(4 - t \cdot 8\right)}{2 + \left(t \cdot t\right) \cdot \left(4 - t \cdot 8\right)}\]

Alternative 9
Error	27.0
Cost	1216

\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{6}\]

Alternative 10
Error	26.1
Cost	64

\[0.5\]

Alternative 11
Error	51.2
Cost	64

\[1\]

Alternative 12
Error	62.0
Cost	64

\[0\]

Alternative 13
Error	63.0
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
\[\leadsto \color{blue}{\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}}\]
Final simplification0.0
\[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

Reproduce

herbie shell --seed 2021042 
(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))))))