Octave 3.8, jcobi/4, as called

Average Error: 46.6 → 0.3

Time: 20.3s

Precision: binary64

Cost: 576

\[i > 0\]

Math

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

↓

\[\frac{0.25}{4 - \frac{1}{i \cdot i}} \]

FPCore

(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 (/ 0.25 (- 4.0 (/ 1.0 (* i i)))))

C

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 0.25 / (4.0 - (1.0 / (i * i)));
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = (((i * i) * (i * i)) / ((2.0d0 * i) * (2.0d0 * i))) / (((2.0d0 * i) * (2.0d0 * i)) - 1.0d0)
end function

↓

real(8) function code(i)
    real(8), intent (in) :: i
    code = 0.25d0 / (4.0d0 - (1.0d0 / (i * i)))
end function

Java

public static double code(double i) {
	return (((i * i) * (i * i)) / ((2.0 * i) * (2.0 * i))) / (((2.0 * i) * (2.0 * i)) - 1.0);
}

↓

public static double code(double i) {
	return 0.25 / (4.0 - (1.0 / (i * i)));
}

Python

def code(i):
	return (((i * i) * (i * i)) / ((2.0 * i) * (2.0 * i))) / (((2.0 * i) * (2.0 * i)) - 1.0)

↓

def code(i):
	return 0.25 / (4.0 - (1.0 / (i * i)))

Julia

function code(i)
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / Float64(Float64(2.0 * i) * Float64(2.0 * i))) / Float64(Float64(Float64(2.0 * i) * Float64(2.0 * i)) - 1.0))
end

↓

function code(i)
	return Float64(0.25 / Float64(4.0 - Float64(1.0 / Float64(i * i))))
end

MATLAB

function tmp = code(i)
	tmp = (((i * i) * (i * i)) / ((2.0 * i) * (2.0 * i))) / (((2.0 * i) * (2.0 * i)) - 1.0);
end

↓

function tmp = code(i)
	tmp = 0.25 / (4.0 - (1.0 / (i * i)));
end

Wolfram

code[i_] := N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[i_] := N[(0.25 / N[(4.0 - N[(1.0 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.6

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

2. Simplified 46.6

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

   [Start] 46.6	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   rational_best-simplify-50 [=>] 46.6	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\color{blue}{i \cdot \left(2 \cdot \left(2 \cdot i\right)\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   rational_best-simplify-50 [=>] 46.6	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{i \cdot \color{blue}{\left(i \cdot \left(2 \cdot 2\right)\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   metadata-eval [=>] 46.6	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{i \cdot \left(i \cdot \color{blue}{4}\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   rational_best-simplify-18 [=>] 46.6	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{i \cdot \left(i \cdot 4\right)}}{\color{blue}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + -1}} \]
   rational_best-simplify-50 [=>] 46.6	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{i \cdot \left(i \cdot 4\right)}}{\color{blue}{i \cdot \left(2 \cdot \left(2 \cdot i\right)\right)} + -1} \]
   rational_best-simplify-50 [=>] 46.6	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{i \cdot \left(i \cdot 4\right)}}{i \cdot \color{blue}{\left(i \cdot \left(2 \cdot 2\right)\right)} + -1} \]
   metadata-eval [=>] 46.6	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{i \cdot \left(i \cdot 4\right)}}{i \cdot \left(i \cdot \color{blue}{4}\right) + -1} \]

3. Applied egg-rr 16.4

   \[\leadsto \frac{\color{blue}{\frac{0.5}{\frac{1}{i}} \cdot \frac{0.5}{\frac{1}{i}}}}{i \cdot \left(i \cdot 4\right) + -1} \]

4. Simplified 16.7

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

   [Start] 16.4	\[ \frac{\frac{0.5}{\frac{1}{i}} \cdot \frac{0.5}{\frac{1}{i}}}{i \cdot \left(i \cdot 4\right) + -1} \]
   rational_best-simplify-79 [=>] 16.8	\[ \frac{\color{blue}{\frac{0.5 \cdot 0.5}{\frac{1}{i} \cdot \frac{1}{i}}}}{i \cdot \left(i \cdot 4\right) + -1} \]
   metadata-eval [=>] 16.8	\[ \frac{\frac{\color{blue}{0.25}}{\frac{1}{i} \cdot \frac{1}{i}}}{i \cdot \left(i \cdot 4\right) + -1} \]
   rational_best-simplify-79 [=>] 16.6	\[ \frac{\frac{0.25}{\color{blue}{\frac{1 \cdot 1}{i \cdot i}}}}{i \cdot \left(i \cdot 4\right) + -1} \]
   metadata-eval [=>] 16.6	\[ \frac{\frac{0.25}{\frac{\color{blue}{1}}{i \cdot i}}}{i \cdot \left(i \cdot 4\right) + -1} \]
   rational_best-simplify-54 [=>] 16.7	\[ \frac{\frac{0.25}{\color{blue}{\frac{\frac{1}{i}}{i}}}}{i \cdot \left(i \cdot 4\right) + -1} \]

5. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\frac{0.25}{4 + \frac{-1}{i \cdot i}} + 0} \]

6. Simplified 0.3

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

   [Start] 0.3	\[ \frac{0.25}{4 + \frac{-1}{i \cdot i}} + 0 \]
   rational_best-simplify-3 [=>] 0.3	\[ \color{blue}{0 + \frac{0.25}{4 + \frac{-1}{i \cdot i}}} \]
   rational_best-simplify-6 [=>] 0.3	\[ \color{blue}{\frac{0.25}{4 + \frac{-1}{i \cdot i}}} \]
   metadata-eval [<=] 0.3	\[ \frac{0.25}{\color{blue}{\left(4 - 0\right)} + \frac{-1}{i \cdot i}} \]
   rational_best-simplify-9 [<=] 0.3	\[ \frac{0.25}{\left(4 - 0\right) + \color{blue}{\left(\frac{-1}{i \cdot i} - 0\right)}} \]
   metadata-eval [<=] 0.3	\[ \frac{0.25}{\left(4 - 0\right) + \left(\frac{-1}{i \cdot i} - \color{blue}{\left(0 - 0\right)}\right)} \]
   rational_best-simplify-51 [<=] 0.3	\[ \frac{0.25}{\left(4 - 0\right) + \color{blue}{\left(0 - \left(0 - \frac{-1}{i \cdot i}\right)\right)}} \]
   rational_best-simplify-14 [<=] 0.3	\[ \frac{0.25}{\left(4 - 0\right) + \left(0 - \color{blue}{\left(-\frac{-1}{i \cdot i}\right)}\right)} \]
   rational_best-simplify-13 [=>] 0.3	\[ \frac{0.25}{\left(4 - 0\right) + \left(0 - \color{blue}{\frac{\frac{-1}{i \cdot i}}{-1}}\right)} \]
   rational_best-simplify-49 [=>] 0.3	\[ \frac{0.25}{\left(4 - 0\right) + \left(0 - \color{blue}{\frac{\frac{-1}{-1}}{i \cdot i}}\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{0.25}{\left(4 - 0\right) + \left(0 - \frac{\color{blue}{1}}{i \cdot i}\right)} \]
   rational_best-simplify-14 [<=] 0.3	\[ \frac{0.25}{\left(4 - 0\right) + \color{blue}{\left(-\frac{1}{i \cdot i}\right)}} \]
   rational_best-simplify-57 [<=] 0.3	\[ \frac{0.25}{\color{blue}{4 - \left(0 + \frac{1}{i \cdot i}\right)}} \]
   rational_best-simplify-6 [=>] 0.3	\[ \frac{0.25}{4 - \color{blue}{\frac{1}{i \cdot i}}} \]

7. Final simplification 0.3

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

Alternatives

Alternative 1
Error	30.9
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (> i 0.0)
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))