Octave 3.8, jcobi/4, as called

Average Error: 47.1 → 0.6

Time: 4.3s

Precision: binary64

Cost: 6788

\[i > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;{i}^{2} \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

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) (* (pow i 2.0) -0.25) 0.0625))

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 = pow(i, 2.0) * -0.25;
	} else {
		tmp = 0.0625;
	}
	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 ** 2.0d0) * (-0.25d0)
    else
        tmp = 0.0625d0
    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 = Math.pow(i, 2.0) * -0.25;
	} else {
		tmp = 0.0625;
	}
	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 = math.pow(i, 2.0) * -0.25
	else:
		tmp = 0.0625
	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((i ^ 2.0) * -0.25);
	else
		tmp = 0.0625;
	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 ^ 2.0) * -0.25;
	else
		tmp = 0.0625;
	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[Power[i, 2.0], $MachinePrecision] * -0.25), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 46.2

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

   2. Simplified 46.3

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

      [Start] 46.2	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-44 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\color{blue}{2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-2 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{2 \cdot \color{blue}{\left(i \cdot \left(2 \cdot i\right)\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-44 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{2 \cdot \color{blue}{\left(2 \cdot \left(i \cdot i\right)\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-2 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{2 \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot 2\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-44 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\color{blue}{\left(i \cdot i\right) \cdot \left(2 \cdot 2\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      metadata-eval [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot \color{blue}{4}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-19 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{\color{blue}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + -1}} \]
      rational_best-simplify-44 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{\color{blue}{2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)} + -1} \]
      rational_best-simplify-2 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{2 \cdot \color{blue}{\left(i \cdot \left(2 \cdot i\right)\right)} + -1} \]
      rational_best-simplify-44 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{2 \cdot \color{blue}{\left(2 \cdot \left(i \cdot i\right)\right)} + -1} \]
      rational_best-simplify-2 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{2 \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot 2\right)} + -1} \]
      rational_best-simplify-44 [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{\color{blue}{\left(i \cdot i\right) \cdot \left(2 \cdot 2\right)} + -1} \]
      metadata-eval [=>] 46.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{\left(i \cdot i\right) \cdot \color{blue}{4} + -1} \]

   3. Taylor expanded in i around 0 0.7

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

   4. Simplified 0.7

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

      [Start] 0.7	\[ -0.25 \cdot {i}^{2} \]
      rational_best-simplify-2 [=>] 0.7	\[ \color{blue}{{i}^{2} \cdot -0.25} \]

   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 48.0

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

      [Start] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-44 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\color{blue}{2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-2 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{2 \cdot \color{blue}{\left(i \cdot \left(2 \cdot i\right)\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-44 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{2 \cdot \color{blue}{\left(2 \cdot \left(i \cdot i\right)\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-2 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{2 \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot 2\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-44 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\color{blue}{\left(i \cdot i\right) \cdot \left(2 \cdot 2\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      metadata-eval [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot \color{blue}{4}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-19 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{\color{blue}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + -1}} \]
      rational_best-simplify-44 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{\color{blue}{2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)} + -1} \]
      rational_best-simplify-2 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{2 \cdot \color{blue}{\left(i \cdot \left(2 \cdot i\right)\right)} + -1} \]
      rational_best-simplify-44 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{2 \cdot \color{blue}{\left(2 \cdot \left(i \cdot i\right)\right)} + -1} \]
      rational_best-simplify-2 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{2 \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot 2\right)} + -1} \]
      rational_best-simplify-44 [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{\color{blue}{\left(i \cdot i\right) \cdot \left(2 \cdot 2\right)} + -1} \]
      metadata-eval [=>] 48.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot i\right) \cdot 4}}{\left(i \cdot i\right) \cdot \color{blue}{4} + -1} \]

   3. Taylor expanded in i around inf 0.6

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;{i}^{2} \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternatives

Alternative 1
Error	23.2
Cost	3012

\[\begin{array}{l} t_0 := \left(i \cdot i\right) \cdot 4\\ t_1 := \left(i \cdot i\right) \cdot \left(i \cdot i\right)\\ t_2 := \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\\ \mathbf{if}\;\frac{\frac{t_1}{t_2}}{t_2 - 1} \leq 0:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 2
Error	31.8
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023092 
(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)))