Average Error: 46.1 → 0.1
Time: 4.4s
Precision: binary64
Cost: 13760
\[i > 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{0.5}{2 + \left|\frac{1}{i}\right|} \cdot \frac{0.5}{2 - \left|\frac{1}{i}\right|}\]
\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{0.5}{2 + \left|\frac{1}{i}\right|} \cdot \frac{0.5}{2 - \left|\frac{1}{i}\right|}
(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
 (* (/ 0.5 (+ 2.0 (fabs (/ 1.0 i)))) (/ 0.5 (- 2.0 (fabs (/ 1.0 i))))))
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 (0.5 / (2.0 + fabs(1.0 / i))) * (0.5 / (2.0 - fabs(1.0 / i)));
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error1.4
Cost40256
\[\frac{\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}}{\sqrt[3]{4 - \frac{1}{i \cdot i}} \cdot \sqrt[3]{4 - \frac{1}{i \cdot i}}} \cdot \frac{\sqrt[3]{0.25}}{\sqrt[3]{4 - \frac{1}{i \cdot i}}}\]
Alternative 2
Error0.2
Cost33088
\[\frac{\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}}{2 + \left|\frac{1}{i}\right|} \cdot \frac{\sqrt[3]{0.25}}{2 - \left|\frac{1}{i}\right|}\]
Alternative 3
Error0.2
Cost20288
\[\frac{\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}}{2 + \frac{1}{i}} \cdot \frac{\sqrt[3]{0.25}}{2 - \frac{1}{i}}\]
Alternative 4
Error0.5
Cost20032
\[\left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}\right) \cdot \frac{\sqrt[3]{0.25}}{4 - \frac{1}{i \cdot i}}\]
Alternative 5
Error46.1
Cost1600
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
Alternative 6
Error0.1
Cost960
\[\frac{0.5}{2 + \frac{1}{i}} \cdot \frac{0.5}{2 - \frac{1}{i}}\]
Alternative 7
Error0.4
Cost576
\[\frac{0.25}{4 - \frac{1}{i \cdot i}}\]
Alternative 8
Error32.0
Cost448
\[0.0625 + \frac{0.015625}{i \cdot i}\]
Alternative 9
Error32.1
Cost320
\[\left(i \cdot i\right) \cdot -0.25\]
Alternative 10
Error31.6
Cost64
\[0.0625\]
Alternative 11
Error58.2
Cost64
\[1\]
Alternative 12
Error46.7
Cost64
\[0\]
Alternative 13
Error62.1
Cost64
\[-1\]

Error

Derivation

  1. Initial program 46.1

    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{1}{i \cdot i}}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt_binary64_4410.5

    \[\leadsto \frac{0.25}{4 - \color{blue}{\sqrt{\frac{1}{i \cdot i}} \cdot \sqrt{\frac{1}{i \cdot i}}}}\]

  5. Applied add-sqr-sqrt_binary64_4410.5

    \[\leadsto \frac{0.25}{\color{blue}{\sqrt{4} \cdot \sqrt{4}} - \sqrt{\frac{1}{i \cdot i}} \cdot \sqrt{\frac{1}{i \cdot i}}}\]

  6. Applied difference-of-squares_binary64_3880.5

    \[\leadsto \frac{0.25}{\color{blue}{\left(\sqrt{4} + \sqrt{\frac{1}{i \cdot i}}\right) \cdot \left(\sqrt{4} - \sqrt{\frac{1}{i \cdot i}}\right)}}\]

  7. Applied add-sqr-sqrt_binary64_4410.5

    \[\leadsto \frac{\color{blue}{\sqrt{0.25} \cdot \sqrt{0.25}}}{\left(\sqrt{4} + \sqrt{\frac{1}{i \cdot i}}\right) \cdot \left(\sqrt{4} - \sqrt{\frac{1}{i \cdot i}}\right)}\]

  8. Applied times-frac_binary64_4250.4

    \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{4} + \sqrt{\frac{1}{i \cdot i}}} \cdot \frac{\sqrt{0.25}}{\sqrt{4} - \sqrt{\frac{1}{i \cdot i}}}}\]

  9. Simplified0.4

    \[\leadsto \color{blue}{\frac{0.5}{2 + \left|\frac{1}{i}\right|}} \cdot \frac{\sqrt{0.25}}{\sqrt{4} - \sqrt{\frac{1}{i \cdot i}}}\]

  10. Simplified0.1

    \[\leadsto \frac{0.5}{2 + \left|\frac{1}{i}\right|} \cdot \color{blue}{\frac{0.5}{2 - \left|\frac{1}{i}\right|}}\]

  11. Simplified0.1

    \[\leadsto \color{blue}{\frac{0.5}{2 + \left|\frac{1}{i}\right|} \cdot \frac{0.5}{2 - \left|\frac{1}{i}\right|}}\]

  12. Final simplification0.1

    \[\leadsto \frac{0.5}{2 + \left|\frac{1}{i}\right|} \cdot \frac{0.5}{2 - \left|\frac{1}{i}\right|}\]

Reproduce

herbie shell --seed 2021022 
(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)))