Average Error: 46.2 → 0.1
Time: 2.3min
Precision: binary64
Cost: 1216
\[i > 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{i}{2}}{i \cdot 2 + 1} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1}\]
\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}
↓
\frac{\frac{i}{2}}{i \cdot 2 + 1} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1}
(FPCore (i)
 :precision binary64
 (/
  (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i)))
  (- (* (* 2.0 i) (* 2.0 i)) 1.0)))
↓
(FPCore (i)
 :precision binary64
 (* (/ (/ i 2.0) (+ (* i 2.0) 1.0)) (/ (/ i 2.0) (- (* i 2.0) 1.0))))
double code(double i) {
	return (((i * i) * (i * i)) / ((2.0 * i) * (2.0 * i))) / (((2.0 * i) * (2.0 * i)) - 1.0);
}
↓
double code(double i) {
	return ((i / 2.0) / ((i * 2.0) + 1.0)) * ((i / 2.0) / ((i * 2.0) - 1.0));
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.1
Cost960
\[\frac{1}{2 + \frac{1}{i}} \cdot \frac{0.25}{2 - \frac{1}{i}}\]
Alternative 2
Error0.1
Cost832
\[\frac{\frac{0.25}{2 + \frac{1}{i}}}{2 - \frac{1}{i}}\]
Alternative 3
Error0.3
Cost576
\[\frac{0.25}{4 - \frac{1}{i \cdot i}}\]
Alternative 4
Error0.4
Cost576
\[\frac{0.25}{4 - \frac{\frac{1}{i}}{i}}\]
Alternative 5
Error0.5
Cost769
\[\begin{array}{l} \mathbf{if}\;i \leq 0.5121179223047105:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array}\]
Alternative 6
Error0.7
Cost641
\[\begin{array}{l} \mathbf{if}\;i \leq 0.5121179223047105:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]
Alternative 7
Error0.7
Cost641
\[\begin{array}{l} \mathbf{if}\;i \leq 0.5121179223047105:\\ \;\;\;\;\left(i \cdot i\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]
Alternative 8
Error16.2
Cost385
\[\begin{array}{l} \mathbf{if}\;i \leq 0.5121179223047105:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]
Alternative 9
Error42.7
Cost385
\[\begin{array}{l} \mathbf{if}\;i \leq 0.5121179223047105:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 10
Error58.2
Cost64
\[1\]

Error

Derivation

  1. Initial program 46.2

    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
  2. Using strategy rm

  3. Applied difference-of-sqr-1_binary64_107146.2

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

  4. Applied times-frac_binary64_110715.5

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

  5. Applied times-frac_binary64_110715.5

    \[\leadsto \color{blue}{\frac{\frac{i \cdot i}{2 \cdot i}}{2 \cdot i + 1} \cdot \frac{\frac{i \cdot i}{2 \cdot i}}{2 \cdot i - 1}}\]

  6. Simplified15.4

    \[\leadsto \color{blue}{\frac{\frac{i}{2}}{i \cdot 2 + 1}} \cdot \frac{\frac{i \cdot i}{2 \cdot i}}{2 \cdot i - 1}\]

  7. Simplified0.1

    \[\leadsto \frac{\frac{i}{2}}{i \cdot 2 + 1} \cdot \color{blue}{\frac{\frac{i}{2}}{i \cdot 2 - 1}}\]

  8. Simplified0.1

    \[\leadsto \color{blue}{\frac{\frac{i}{2}}{i \cdot 2 + 1} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1}}\]

  9. Final simplification0.1

    \[\leadsto \frac{\frac{i}{2}}{i \cdot 2 + 1} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1}\]

Reproduce

herbie shell --seed 2021014 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))