Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.5% → 100.0%

Time: 6.8s

Alternatives: 5

Speedup: 25.0×

Specification

\[i > 0\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\\ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0}}{t\_0 - 1} \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* 2.0 i) (* 2.0 i))))
   (/ (/ (* (* i i) (* i i)) t_0) (- t_0 1.0))))

C

double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (2.0d0 * i) * (2.0d0 * i)
    code = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0d0)
end function

Java

public static double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Python

def code(i):
	t_0 = (2.0 * i) * (2.0 * i)
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0)

Julia

function code(i)
	t_0 = Float64(Float64(2.0 * i) * Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(i)
	t_0 = (2.0 * i) * (2.0 * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
end

Wolfram

code[i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 27.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\\ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0}}{t\_0 - 1} \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* 2.0 i) (* 2.0 i))))
   (/ (/ (* (* i i) (* i i)) t_0) (- t_0 1.0))))

C

double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (2.0d0 * i) * (2.0d0 * i)
    code = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0d0)
end function

Java

public static double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Python

def code(i):
	t_0 = (2.0 * i) * (2.0 * i)
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0)

Julia

function code(i)
	t_0 = Float64(Float64(2.0 * i) * Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(i)
	t_0 = (2.0 * i) * (2.0 * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
end

Wolfram

code[i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 10000:\\ \;\;\;\;\frac{{i}^{2} \cdot 0.25}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 10000.0)
   (/ (* (pow i 2.0) 0.25) (+ (* (* i 2.0) (* i 2.0)) -1.0))
   (+ 0.0625 (/ (/ 0.015625 i) i))))

C

double code(double i) {
	double tmp;
	if (i <= 10000.0) {
		tmp = (pow(i, 2.0) * 0.25) / (((i * 2.0) * (i * 2.0)) + -1.0);
	} else {
		tmp = 0.0625 + ((0.015625 / i) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 10000.0d0) then
        tmp = ((i ** 2.0d0) * 0.25d0) / (((i * 2.0d0) * (i * 2.0d0)) + (-1.0d0))
    else
        tmp = 0.0625d0 + ((0.015625d0 / i) / i)
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 10000.0) {
		tmp = (Math.pow(i, 2.0) * 0.25) / (((i * 2.0) * (i * 2.0)) + -1.0);
	} else {
		tmp = 0.0625 + ((0.015625 / i) / i);
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 10000.0:
		tmp = (math.pow(i, 2.0) * 0.25) / (((i * 2.0) * (i * 2.0)) + -1.0)
	else:
		tmp = 0.0625 + ((0.015625 / i) / i)
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 10000.0)
		tmp = Float64(Float64((i ^ 2.0) * 0.25) / Float64(Float64(Float64(i * 2.0) * Float64(i * 2.0)) + -1.0));
	else
		tmp = Float64(0.0625 + Float64(Float64(0.015625 / i) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 10000.0)
		tmp = ((i ^ 2.0) * 0.25) / (((i * 2.0) * (i * 2.0)) + -1.0);
	else
		tmp = 0.0625 + ((0.015625 / i) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 10000.0], N[(N[(N[Power[i, 2.0], $MachinePrecision] * 0.25), $MachinePrecision] / N[(N[(N[(i * 2.0), $MachinePrecision] * N[(i * 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(N[(0.015625 / i), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1e4

   1. Initial program 27.9%

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

   3. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot 0.25}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot 0.25}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   if 1e4 < i

   1. Initial program 27.4%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. sqr-neg 27.4%

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

      2. sqr-neg 27.4%

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

      3. swap-sqr 27.4%

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

      4. sqr-neg 27.4%

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

      5. swap-sqr 27.4%

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

      6. associate-/r* 27.0%

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

      7. sqr-neg 27.0%

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

      8. associate-*l* 26.8%

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

      9. sqr-neg 26.8%

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

      10. cube-unmult 26.8%

          \[\leadsto \frac{i \cdot \color{blue}{{i}^{3}}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot \left(-i\right)\right) \cdot \left(2 \cdot \left(-i\right)\right) - 1\right)} \]

   3. Simplified 26.8%

      \[\leadsto \color{blue}{\frac{i \cdot {i}^{3}}{\left(i \cdot \left(i \cdot 4\right)\right) \cdot \mathsf{fma}\left(i \cdot 4, i, -1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 100.0%

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-div 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto 0.0625 + \frac{\color{blue}{0.125}}{\sqrt{{i}^{2}}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      4. sqrt-pow1 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{\color{blue}{{i}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      5. metadata-eval 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{{i}^{\color{blue}{1}}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      6. pow1 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{\color{blue}{i}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      7. sqrt-div 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \color{blue}{\frac{\sqrt{0.015625}}{\sqrt{{i}^{2}}}} \]

      8. metadata-eval 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{\color{blue}{0.125}}{\sqrt{{i}^{2}}} \]

      9. sqrt-pow1 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{0.125}{\color{blue}{{i}^{\left(\frac{2}{2}\right)}}} \]

      10. metadata-eval 100.0%

          \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{0.125}{{i}^{\color{blue}{1}}} \]

      11. pow1 100.0%

          \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{0.125}{\color{blue}{i}} \]

   9. Applied egg-rr 100.0%

      \[\leadsto 0.0625 + \color{blue}{\frac{0.125}{i} \cdot \frac{0.125}{i}} \]
   10. Step-by-step derivation

       1. associate-*l/ 100.0%

          \[\leadsto 0.0625 + \color{blue}{\frac{0.125 \cdot \frac{0.125}{i}}{i}} \]

       2. associate-*r/ 100.0%

          \[\leadsto 0.0625 + \frac{\color{blue}{\frac{0.125 \cdot 0.125}{i}}}{i} \]

       3. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{\frac{0.015625}{i}}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 10000:\\ \;\;\;\;\frac{{i}^{2} \cdot 0.25}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 4000:\\ \;\;\;\;0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 4000.0)
   (* 0.25 (* i (/ i (fma i (* i 4.0) -1.0))))
   (+ 0.0625 (/ (/ 0.015625 i) i))))

C

double code(double i) {
	double tmp;
	if (i <= 4000.0) {
		tmp = 0.25 * (i * (i / fma(i, (i * 4.0), -1.0)));
	} else {
		tmp = 0.0625 + ((0.015625 / i) / i);
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 4000.0)
		tmp = Float64(0.25 * Float64(i * Float64(i / fma(i, Float64(i * 4.0), -1.0))));
	else
		tmp = Float64(0.0625 + Float64(Float64(0.015625 / i) / i));
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 4000.0], N[(0.25 * N[(i * N[(i / N[(i * N[(i * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(N[(0.015625 / i), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 4e3

   1. Initial program 27.9%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
   4. Add Preprocessing

   if 4e3 < i

   1. Initial program 27.4%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. sqr-neg 27.4%

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

      2. sqr-neg 27.4%

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

      3. swap-sqr 27.4%

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

      4. sqr-neg 27.4%

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

      5. swap-sqr 27.4%

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

      6. associate-/r* 27.0%

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

      7. sqr-neg 27.0%

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

      8. associate-*l* 26.8%

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

      9. sqr-neg 26.8%

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

      10. cube-unmult 26.8%

          \[\leadsto \frac{i \cdot \color{blue}{{i}^{3}}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot \left(-i\right)\right) \cdot \left(2 \cdot \left(-i\right)\right) - 1\right)} \]

   3. Simplified 26.8%

      \[\leadsto \color{blue}{\frac{i \cdot {i}^{3}}{\left(i \cdot \left(i \cdot 4\right)\right) \cdot \mathsf{fma}\left(i \cdot 4, i, -1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 100.0%

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-div 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto 0.0625 + \frac{\color{blue}{0.125}}{\sqrt{{i}^{2}}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      4. sqrt-pow1 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{\color{blue}{{i}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      5. metadata-eval 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{{i}^{\color{blue}{1}}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      6. pow1 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{\color{blue}{i}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      7. sqrt-div 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \color{blue}{\frac{\sqrt{0.015625}}{\sqrt{{i}^{2}}}} \]

      8. metadata-eval 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{\color{blue}{0.125}}{\sqrt{{i}^{2}}} \]

      9. sqrt-pow1 100.0%

         \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{0.125}{\color{blue}{{i}^{\left(\frac{2}{2}\right)}}} \]

      10. metadata-eval 100.0%

          \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{0.125}{{i}^{\color{blue}{1}}} \]

      11. pow1 100.0%

          \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{0.125}{\color{blue}{i}} \]

   9. Applied egg-rr 100.0%

      \[\leadsto 0.0625 + \color{blue}{\frac{0.125}{i} \cdot \frac{0.125}{i}} \]
   10. Step-by-step derivation

       1. associate-*l/ 100.0%

          \[\leadsto 0.0625 + \color{blue}{\frac{0.125 \cdot \frac{0.125}{i}}{i}} \]

       2. associate-*r/ 100.0%

          \[\leadsto 0.0625 + \frac{\color{blue}{\frac{0.125 \cdot 0.125}{i}}}{i} \]

       3. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{\frac{0.015625}{i}}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 4000:\\ \;\;\;\;0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5) (* 0.25 (* i (- i))) (+ 0.0625 (/ (/ 0.015625 i) i))))

C

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

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.5d0) then
        tmp = 0.25d0 * (i * -i)
    else
        tmp = 0.0625d0 + ((0.015625d0 / i) / i)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(0.25 * Float64(i * Float64(-i)));
	else
		tmp = Float64(0.0625 + Float64(Float64(0.015625 / i) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = 0.25 * (i * -i);
	else
		tmp = 0.0625 + ((0.015625 / i) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(0.25 * N[(i * (-i)), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(N[(0.015625 / i), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 26.9%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 98.4%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 98.4%

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

   7. Simplified 98.4%

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

   if 0.5 < i

   1. Initial program 28.7%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. sqr-neg 28.7%

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

      2. sqr-neg 28.7%

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

      3. swap-sqr 28.7%

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

      4. sqr-neg 28.7%

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

      5. swap-sqr 28.7%

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

      6. associate-/r* 28.3%

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

      7. sqr-neg 28.3%

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

      8. associate-*l* 28.1%

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

      9. sqr-neg 28.1%

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

      10. cube-unmult 28.1%

          \[\leadsto \frac{i \cdot \color{blue}{{i}^{3}}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot \left(-i\right)\right) \cdot \left(2 \cdot \left(-i\right)\right) - 1\right)} \]

   3. Simplified 28.1%

      \[\leadsto \color{blue}{\frac{i \cdot {i}^{3}}{\left(i \cdot \left(i \cdot 4\right)\right) \cdot \mathsf{fma}\left(i \cdot 4, i, -1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 99.4%

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   7. Simplified 99.4%

      \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 99.4%

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

      2. sqrt-div 99.4%

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

      3. metadata-eval 99.4%

         \[\leadsto 0.0625 + \frac{\color{blue}{0.125}}{\sqrt{{i}^{2}}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      4. sqrt-pow1 99.4%

         \[\leadsto 0.0625 + \frac{0.125}{\color{blue}{{i}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      5. metadata-eval 99.4%

         \[\leadsto 0.0625 + \frac{0.125}{{i}^{\color{blue}{1}}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      6. pow1 99.4%

         \[\leadsto 0.0625 + \frac{0.125}{\color{blue}{i}} \cdot \sqrt{\frac{0.015625}{{i}^{2}}} \]

      7. sqrt-div 99.4%

         \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \color{blue}{\frac{\sqrt{0.015625}}{\sqrt{{i}^{2}}}} \]

      8. metadata-eval 99.4%

         \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{\color{blue}{0.125}}{\sqrt{{i}^{2}}} \]

      9. sqrt-pow1 99.4%

         \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{0.125}{\color{blue}{{i}^{\left(\frac{2}{2}\right)}}} \]

      10. metadata-eval 99.4%

          \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{0.125}{{i}^{\color{blue}{1}}} \]

      11. pow1 99.4%

          \[\leadsto 0.0625 + \frac{0.125}{i} \cdot \frac{0.125}{\color{blue}{i}} \]

   9. Applied egg-rr 99.4%

      \[\leadsto 0.0625 + \color{blue}{\frac{0.125}{i} \cdot \frac{0.125}{i}} \]
   10. Step-by-step derivation

       1. associate-*l/ 99.4%

          \[\leadsto 0.0625 + \color{blue}{\frac{0.125 \cdot \frac{0.125}{i}}{i}} \]

       2. associate-*r/ 99.4%

          \[\leadsto 0.0625 + \frac{\color{blue}{\frac{0.125 \cdot 0.125}{i}}}{i} \]

       3. metadata-eval 99.4%

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

   11. Applied egg-rr 99.4%

       \[\leadsto 0.0625 + \color{blue}{\frac{\frac{0.015625}{i}}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

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

Alternative 4: 98.8% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (i) :precision binary64 (if (<= i 0.5) (* 0.25 (* i (- i))) 0.0625))

C

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

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.5d0) then
        tmp = 0.25d0 * (i * -i)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = 0.25 * (i * -i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = 0.25 * (i * -i)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(0.25 * Float64(i * Float64(-i)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = 0.25 * (i * -i);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(0.25 * N[(i * (-i)), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 26.9%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 98.4%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 98.4%

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

   7. Simplified 98.4%

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

   if 0.5 < i

   1. Initial program 28.7%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. sqr-neg 28.7%

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

      2. sqr-neg 28.7%

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

      3. swap-sqr 28.7%

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

      4. sqr-neg 28.7%

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

      5. swap-sqr 28.7%

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

      6. associate-/r* 28.3%

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

      7. sqr-neg 28.3%

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

      8. associate-*l* 28.1%

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

      9. sqr-neg 28.1%

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

      10. cube-unmult 28.1%

          \[\leadsto \frac{i \cdot \color{blue}{{i}^{3}}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot \left(-i\right)\right) \cdot \left(2 \cdot \left(-i\right)\right) - 1\right)} \]

   3. Simplified 28.1%

      \[\leadsto \color{blue}{\frac{i \cdot {i}^{3}}{\left(i \cdot \left(i \cdot 4\right)\right) \cdot \mathsf{fma}\left(i \cdot 4, i, -1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 98.3%

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

4. Final simplification 98.4%

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

Alternative 5: 50.4% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (i) :precision binary64 0.0625)

C

double code(double i) {
	return 0.0625;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

public static double code(double i) {
	return 0.0625;
}

Python

def code(i):
	return 0.0625

Julia

function code(i)
	return 0.0625
end

MATLAB

function tmp = code(i)
	tmp = 0.0625;
end

Wolfram

code[i_] := 0.0625

Derivation

1. Initial program 27.7%

   \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
2. Step-by-step derivation

   1. sqr-neg 27.7%

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

   2. sqr-neg 27.7%

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

   3. swap-sqr 27.7%

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

   4. sqr-neg 27.7%

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

   5. swap-sqr 27.7%

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

   6. associate-/r* 27.5%

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

   7. sqr-neg 27.5%

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

   8. associate-*l* 27.4%

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

   9. sqr-neg 27.4%

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

   10. cube-unmult 27.3%

       \[\leadsto \frac{i \cdot \color{blue}{{i}^{3}}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot \left(-i\right)\right) \cdot \left(2 \cdot \left(-i\right)\right) - 1\right)} \]

3. Simplified 27.3%

   \[\leadsto \color{blue}{\frac{i \cdot {i}^{3}}{\left(i \cdot \left(i \cdot 4\right)\right) \cdot \mathsf{fma}\left(i \cdot 4, i, -1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in i around inf 44.9%

   \[\leadsto \color{blue}{0.0625} \]

6. Final simplification 44.9%

   \[\leadsto 0.0625 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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)))