Octave 3.8, jcobi/4, as called

Average Accuracy: 26.5% → 99.3%

Time: 3.6s

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 28.9%

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

   2. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{i}{{i}^{3}}}} \]
      Proof

      [Start] 28.9	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      times-frac [=>] 99.9	\[ \frac{\color{blue}{\frac{i \cdot i}{2 \cdot i} \cdot \frac{i \cdot i}{2 \cdot i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      associate-/l* [=>] 99.7	\[ \color{blue}{\frac{\frac{i \cdot i}{2 \cdot i}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{\frac{i \cdot i}{2 \cdot i}}}} \]
      associate-/l* [=>] 99.8	\[ \frac{\color{blue}{\frac{i}{\frac{2 \cdot i}{i}}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{\frac{i \cdot i}{2 \cdot i}}} \]
      associate-/l/ [=>] 99.8	\[ \color{blue}{\frac{i}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{\frac{i \cdot i}{2 \cdot i}} \cdot \frac{2 \cdot i}{i}}} \]
      associate-/r/ [=>] 98.9	\[ \frac{i}{\color{blue}{\left(\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i} \cdot \left(2 \cdot i\right)\right)} \cdot \frac{2 \cdot i}{i}} \]
      associate-/l* [=>] 98.9	\[ \frac{i}{\left(\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i} \cdot \left(2 \cdot i\right)\right) \cdot \color{blue}{\frac{2}{\frac{i}{i}}}} \]
      *-inverses [=>] 98.9	\[ \frac{i}{\left(\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i} \cdot \left(2 \cdot i\right)\right) \cdot \frac{2}{\color{blue}{1}}} \]
      metadata-eval [=>] 98.9	\[ \frac{i}{\left(\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i} \cdot \left(2 \cdot i\right)\right) \cdot \color{blue}{2}} \]
      associate-*l* [=>] 98.9	\[ \frac{i}{\color{blue}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i} \cdot \left(\left(2 \cdot i\right) \cdot 2\right)}} \]
      *-commutative [<=] 98.9	\[ \frac{i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot 2\right) \cdot \frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}}} \]
      associate-/r* [=>] 99.0	\[ \color{blue}{\frac{\frac{i}{\left(2 \cdot i\right) \cdot 2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}}} \]

   3. Taylor expanded in i around 0 98.9%

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

   4. Simplified 98.9%

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

      [Start] 98.9	\[ -0.25 \cdot {i}^{2} \]
      *-commutative [=>] 98.9	\[ \color{blue}{{i}^{2} \cdot -0.25} \]
      unpow2 [=>] 98.9	\[ \color{blue}{\left(i \cdot i\right)} \cdot -0.25 \]

   if 0.5 < i

   1. Initial program 24.1%

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

   2. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{i}{{i}^{3}}}} \]
      Proof

      [Start] 24.1	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      times-frac [=>] 48.7	\[ \frac{\color{blue}{\frac{i \cdot i}{2 \cdot i} \cdot \frac{i \cdot i}{2 \cdot i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      associate-/l* [=>] 48.7	\[ \color{blue}{\frac{\frac{i \cdot i}{2 \cdot i}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{\frac{i \cdot i}{2 \cdot i}}}} \]
      associate-/l* [=>] 48.8	\[ \frac{\color{blue}{\frac{i}{\frac{2 \cdot i}{i}}}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{\frac{i \cdot i}{2 \cdot i}}} \]
      associate-/l/ [=>] 48.8	\[ \color{blue}{\frac{i}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{\frac{i \cdot i}{2 \cdot i}} \cdot \frac{2 \cdot i}{i}}} \]
      associate-/r/ [=>] 48.8	\[ \frac{i}{\color{blue}{\left(\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i} \cdot \left(2 \cdot i\right)\right)} \cdot \frac{2 \cdot i}{i}} \]
      associate-/l* [=>] 48.8	\[ \frac{i}{\left(\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i} \cdot \left(2 \cdot i\right)\right) \cdot \color{blue}{\frac{2}{\frac{i}{i}}}} \]
      *-inverses [=>] 48.8	\[ \frac{i}{\left(\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i} \cdot \left(2 \cdot i\right)\right) \cdot \frac{2}{\color{blue}{1}}} \]
      metadata-eval [=>] 48.8	\[ \frac{i}{\left(\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i} \cdot \left(2 \cdot i\right)\right) \cdot \color{blue}{2}} \]
      associate-*l* [=>] 48.8	\[ \frac{i}{\color{blue}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i} \cdot \left(\left(2 \cdot i\right) \cdot 2\right)}} \]
      *-commutative [<=] 48.8	\[ \frac{i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot 2\right) \cdot \frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}}} \]
      associate-/r* [=>] 48.8	\[ \color{blue}{\frac{\frac{i}{\left(2 \cdot i\right) \cdot 2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{i \cdot i}}} \]

   3. Taylor expanded in i around inf 99.7%

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

   4. Simplified 99.7%

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

      [Start] 99.7	\[ 0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}} \]
      associate-*r/ [=>] 99.7	\[ 0.0625 + \color{blue}{\frac{0.015625 \cdot 1}{{i}^{2}}} \]
      metadata-eval [=>] 99.7	\[ 0.0625 + \frac{\color{blue}{0.015625}}{{i}^{2}} \]
      unpow2 [=>] 99.7	\[ 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	576

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

Alternative 2
Accuracy	99.0%
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
Accuracy	51.1%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023146 
(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)))