Octave 3.8, jcobi/4, as called

Average Error: 47.0 → 0.3

Time: 2.9s

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 47.0

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

2. Simplified 0.3

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

   [Start] 47.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   associate-*l/ [<=] 31.1	\[ \frac{\color{blue}{\frac{i \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)} \cdot \left(i \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   associate-/l* [=>] 31.4	\[ \color{blue}{\frac{\frac{i \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}}} \]
   associate-*l* [=>] 31.4	\[ \frac{\frac{i \cdot i}{\color{blue}{2 \cdot \left(i \cdot \left(2 \cdot i\right)\right)}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
   *-commutative [=>] 31.4	\[ \frac{\frac{i \cdot i}{\color{blue}{\left(i \cdot \left(2 \cdot i\right)\right) \cdot 2}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
   associate-/r* [=>] 31.4	\[ \frac{\color{blue}{\frac{\frac{i \cdot i}{i \cdot \left(2 \cdot i\right)}}{2}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
   *-commutative [=>] 31.4	\[ \frac{\frac{\frac{i \cdot i}{i \cdot \color{blue}{\left(i \cdot 2\right)}}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
   associate-*r* [=>] 31.4	\[ \frac{\frac{\frac{i \cdot i}{\color{blue}{\left(i \cdot i\right) \cdot 2}}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
   associate-/r* [=>] 31.4	\[ \frac{\frac{\color{blue}{\frac{\frac{i \cdot i}{i \cdot i}}{2}}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
   *-inverses [=>] 16.4	\[ \frac{\frac{\frac{\color{blue}{1}}{2}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
   metadata-eval [=>] 16.4	\[ \frac{\frac{\color{blue}{0.5}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
   metadata-eval [=>] 16.4	\[ \frac{\color{blue}{0.25}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
   *-inverses [<=] 16.4	\[ \frac{0.25}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - \color{blue}{\frac{i}{i}}}{i \cdot i}} \]
   div-sub [=>] 31.4	\[ \frac{0.25}{\color{blue}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}{i \cdot i} - \frac{\frac{i}{i}}{i \cdot i}}} \]

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 3
Error	31.5
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023053 
(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)))