Octave 3.8, jcobi/4, as called

Average Error: 72.76% → 0.33%

Time: 5.2s

Precision: binary64

Cost: 640

\[i > 0\]

Math

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

↓

\[\frac{-i}{\frac{4}{i} + i \cdot -16} \]

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 (/ (- i) (+ (/ 4.0 i) (* i -16.0))))

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 -i / ((4.0 / i) + (i * -16.0));
}

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 = -i / ((4.0d0 / i) + (i * (-16.0d0)))
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 -i / ((4.0 / i) + (i * -16.0));
}

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 -i / ((4.0 / i) + (i * -16.0))

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(Float64(-i) / Float64(Float64(4.0 / i) + Float64(i * -16.0)))
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 = -i / ((4.0 / i) + (i * -16.0));
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[((-i) / N[(N[(4.0 / i), $MachinePrecision] + N[(i * -16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.76

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

2. Simplified 49.32

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

   [Start] 72.76	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   associate-/l* [=>] 49.27	\[ \frac{\color{blue}{\frac{i \cdot i}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}{i \cdot i}}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   swap-sqr [=>] 49.32	\[ \frac{\frac{i \cdot i}{\frac{\color{blue}{\left(2 \cdot 2\right) \cdot \left(i \cdot i\right)}}{i \cdot i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   metadata-eval [=>] 49.32	\[ \frac{\frac{i \cdot i}{\frac{\color{blue}{4} \cdot \left(i \cdot i\right)}{i \cdot i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   sub-neg [=>] 49.32	\[ \frac{\frac{i \cdot i}{\frac{4 \cdot \left(i \cdot i\right)}{i \cdot i}}}{\color{blue}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + \left(-1\right)}} \]
   swap-sqr [=>] 49.32	\[ \frac{\frac{i \cdot i}{\frac{4 \cdot \left(i \cdot i\right)}{i \cdot i}}}{\color{blue}{\left(2 \cdot 2\right) \cdot \left(i \cdot i\right)} + \left(-1\right)} \]
   metadata-eval [=>] 49.32	\[ \frac{\frac{i \cdot i}{\frac{4 \cdot \left(i \cdot i\right)}{i \cdot i}}}{\color{blue}{4} \cdot \left(i \cdot i\right) + \left(-1\right)} \]
   metadata-eval [=>] 49.32	\[ \frac{\frac{i \cdot i}{\frac{4 \cdot \left(i \cdot i\right)}{i \cdot i}}}{4 \cdot \left(i \cdot i\right) + \color{blue}{-1}} \]

3. Applied egg-rr 24.66

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

4. Simplified 24.66

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

   [Start] 24.66	\[ -\frac{i}{\left(1 + \left(i \cdot i\right) \cdot -4\right) \cdot \frac{4}{i}} \]
   distribute-neg-frac [=>] 24.66	\[ \color{blue}{\frac{-i}{\left(1 + \left(i \cdot i\right) \cdot -4\right) \cdot \frac{4}{i}}} \]
   *-commutative [=>] 24.66	\[ \frac{-i}{\color{blue}{\frac{4}{i} \cdot \left(1 + \left(i \cdot i\right) \cdot -4\right)}} \]
   associate-*l* [=>] 24.66	\[ \frac{-i}{\frac{4}{i} \cdot \left(1 + \color{blue}{i \cdot \left(i \cdot -4\right)}\right)} \]

5. Taylor expanded in i around 0 0.33

   \[\leadsto \frac{-i}{\color{blue}{4 \cdot \frac{1}{i} + -16 \cdot i}} \]

6. Simplified 0.33

   \[\leadsto \frac{-i}{\color{blue}{\frac{4}{i} + i \cdot -16}} \]
   Proof

   [Start] 0.33	\[ \frac{-i}{4 \cdot \frac{1}{i} + -16 \cdot i} \]
   associate-*r/ [=>] 0.33	\[ \frac{-i}{\color{blue}{\frac{4 \cdot 1}{i}} + -16 \cdot i} \]
   metadata-eval [=>] 0.33	\[ \frac{-i}{\frac{\color{blue}{4}}{i} + -16 \cdot i} \]
   *-commutative [=>] 0.33	\[ \frac{-i}{\frac{4}{i} + \color{blue}{i \cdot -16}} \]

7. Final simplification 0.33

   \[\leadsto \frac{-i}{\frac{4}{i} + i \cdot -16} \]

Alternatives

Alternative 1
Error	0.75%
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	0.5%
Cost	576

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

Alternative 3
Error	0.97%
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 4
Error	48.61%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023102 
(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)))