Average Error: 46.9 → 0.4
Time: 1.8s
Precision: binary64
\[i > 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}\right) \cdot \frac{\sqrt[3]{0.25}}{4 + \frac{-1}{i \cdot i}}\]
\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}
↓
\left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}\right) \cdot \frac{\sqrt[3]{0.25}}{4 + \frac{-1}{i \cdot i}}
(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
 (* (* (cbrt 0.25) (cbrt 0.25)) (/ (cbrt 0.25) (+ 4.0 (/ -1.0 (* i 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 (cbrt(0.25) * cbrt(0.25)) * (cbrt(0.25) / (4.0 + (-1.0 / (i * i))));
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.9

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

  2. Simplified0.3

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

  4. Applied *-un-lft-identity_binary640.3

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

  5. Applied add-cube-cbrt_binary640.3

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}\right) \cdot \sqrt[3]{0.25}}}{1 \cdot \left(4 + \frac{-1}{i \cdot i}\right)}\]

  6. Applied times-frac_binary640.4

    \[\leadsto \color{blue}{\frac{\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}}{1} \cdot \frac{\sqrt[3]{0.25}}{4 + \frac{-1}{i \cdot i}}}\]

  7. Simplified0.4

    \[\leadsto \color{blue}{\left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}\right)} \cdot \frac{\sqrt[3]{0.25}}{4 + \frac{-1}{i \cdot i}}\]

  8. Final simplification0.4

    \[\leadsto \left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}\right) \cdot \frac{\sqrt[3]{0.25}}{4 + \frac{-1}{i \cdot i}}\]

Reproduce

herbie shell --seed 2020253 
(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)))