Octave 3.8, jcobi/4, as called

Average Error: 46.9 → 0.5

Time: 2.7s

Precision: binary64

Cost: 580

\[i > 0\]

Math

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

↓

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

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
 (if (<= i 0.5) (* (* i i) -0.25) (+ 0.0625 (/ 0.015625 (* 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) {
	double tmp;
	if (i <= 0.5) {
		tmp = (i * i) * -0.25;
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (i <= 0.5d0) then
        tmp = (i * i) * (-0.25d0)
    else
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    end if
    code = tmp
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) {
	double tmp;
	if (i <= 0.5) {
		tmp = (i * i) * -0.25;
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

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):
	tmp = 0
	if i <= 0.5:
		tmp = (i * i) * -0.25
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

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)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(Float64(i * i) * -0.25);
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
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_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = (i * i) * -0.25;
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
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_] := If[LessEqual[i, 0.5], N[(N[(i * i), $MachinePrecision] * -0.25), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 45.8

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

   2. Simplified 0.7

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

      [Start] 45.8	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      associate-*l/ [<=] 29.8	\[ \frac{\color{blue}{\frac{i \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)} \cdot \left(i \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      associate-/l* [=>] 30.5	\[ \color{blue}{\frac{\frac{i \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}}} \]
      associate-*l* [=>] 30.5	\[ \frac{\frac{i \cdot i}{\color{blue}{2 \cdot \left(i \cdot \left(2 \cdot i\right)\right)}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      *-commutative [=>] 30.5	\[ \frac{\frac{i \cdot i}{\color{blue}{\left(i \cdot \left(2 \cdot i\right)\right) \cdot 2}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      associate-/r* [=>] 30.5	\[ \frac{\color{blue}{\frac{\frac{i \cdot i}{i \cdot \left(2 \cdot i\right)}}{2}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      *-commutative [=>] 30.5	\[ \frac{\frac{\frac{i \cdot i}{i \cdot \color{blue}{\left(i \cdot 2\right)}}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      associate-*r* [=>] 30.6	\[ \frac{\frac{\frac{i \cdot i}{\color{blue}{\left(i \cdot i\right) \cdot 2}}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      associate-/r* [=>] 30.6	\[ \frac{\frac{\color{blue}{\frac{\frac{i \cdot i}{i \cdot i}}{2}}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      *-inverses [=>] 0.7	\[ \frac{\frac{\frac{\color{blue}{1}}{2}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      metadata-eval [=>] 0.7	\[ \frac{\frac{\color{blue}{0.5}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      metadata-eval [=>] 0.7	\[ \frac{\color{blue}{0.25}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      *-inverses [<=] 0.7	\[ \frac{0.25}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - \color{blue}{\frac{i}{i}}}{i \cdot i}} \]
      div-sub [=>] 30.6	\[ \frac{0.25}{\color{blue}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}{i \cdot i} - \frac{\frac{i}{i}}{i \cdot i}}} \]

   3. Taylor expanded in i around 0 0.6

      \[\leadsto \color{blue}{-0.25 \cdot {i}^{2}} \]

   4. Simplified 0.6

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

      [Start] 0.6	\[ -0.25 \cdot {i}^{2} \]
      *-commutative [=>] 0.6	\[ \color{blue}{{i}^{2} \cdot -0.25} \]
      unpow2 [=>] 0.6	\[ \color{blue}{\left(i \cdot i\right)} \cdot -0.25 \]

   if 0.5 < i

   1. Initial program 48.0

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

   2. Simplified 0.0

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

      [Start] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      associate-*l/ [<=] 32.2	\[ \frac{\color{blue}{\frac{i \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)} \cdot \left(i \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      associate-/l* [=>] 32.2	\[ \color{blue}{\frac{\frac{i \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}}} \]
      associate-*l* [=>] 32.2	\[ \frac{\frac{i \cdot i}{\color{blue}{2 \cdot \left(i \cdot \left(2 \cdot i\right)\right)}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      *-commutative [=>] 32.2	\[ \frac{\frac{i \cdot i}{\color{blue}{\left(i \cdot \left(2 \cdot i\right)\right) \cdot 2}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      associate-/r* [=>] 32.2	\[ \frac{\color{blue}{\frac{\frac{i \cdot i}{i \cdot \left(2 \cdot i\right)}}{2}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      *-commutative [=>] 32.2	\[ \frac{\frac{\frac{i \cdot i}{i \cdot \color{blue}{\left(i \cdot 2\right)}}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      associate-*r* [=>] 32.2	\[ \frac{\frac{\frac{i \cdot i}{\color{blue}{\left(i \cdot i\right) \cdot 2}}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      associate-/r* [=>] 32.2	\[ \frac{\frac{\color{blue}{\frac{\frac{i \cdot i}{i \cdot i}}{2}}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      *-inverses [=>] 32.2	\[ \frac{\frac{\frac{\color{blue}{1}}{2}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      metadata-eval [=>] 32.2	\[ \frac{\frac{\color{blue}{0.5}}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      metadata-eval [=>] 32.2	\[ \frac{\color{blue}{0.25}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}} \]
      *-inverses [<=] 32.2	\[ \frac{0.25}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - \color{blue}{\frac{i}{i}}}{i \cdot i}} \]
      div-sub [=>] 32.2	\[ \frac{0.25}{\color{blue}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}{i \cdot i} - \frac{\frac{i}{i}}{i \cdot i}}} \]

   3. Taylor expanded in i around inf 0.4

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]

   4. Simplified 0.4

      \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
      Proof

      [Start] 0.4	\[ 0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}} \]
      associate-*r/ [=>] 0.4	\[ 0.0625 + \color{blue}{\frac{0.015625 \cdot 1}{{i}^{2}}} \]
      metadata-eval [=>] 0.4	\[ 0.0625 + \frac{\color{blue}{0.015625}}{{i}^{2}} \]
      unpow2 [=>] 0.4	\[ 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.4
Cost	576

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

Alternative 2
Error	0.7
Cost	452

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

Alternative 3
Error	31.7
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023018 
(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)))