Octave 3.8, jcobi/4, as called

Average Error: 46.6 → 0.4

Time: 2.8s

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 0.4

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

3. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;i \leq 0.3744150222762028:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;i \leq 0.3744150222762028:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 3
Error	31.5
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022225 
(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)))