Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.6% → 99.2%

Time: 3.1s

Alternatives: 4

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: 27.6% 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: 99.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

MATLAB

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_] := If[LessEqual[i, 0.5], N[(i * N[(i * -0.25), $MachinePrecision]), $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 24.9%

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

      1. times-frac 99.9%

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

      2. associate-/l* 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-/l/ 99.7%

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

      5. associate-/r/ 98.4%

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

      6. associate-/l* 98.4%

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

      7. *-inverses 98.4%

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

      8. metadata-eval 98.4%

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

      9. associate-*l* 98.4%

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

      10. *-commutative 98.4%

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

      11. associate-/r* 98.5%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in i around 0 99.3%

      \[\leadsto \color{blue}{-0.25 \cdot {i}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l* 99.3%

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

   6. Simplified 99.3%

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

   if 0.5 < i

   1. Initial program 25.9%

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

      1. times-frac 48.8%

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

      2. associate-/l* 48.7%

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

      3. associate-/l* 48.8%

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

      4. associate-/l/ 48.8%

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

      5. associate-/r/ 48.8%

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

      6. associate-/l* 48.8%

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

      7. *-inverses 48.8%

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

      8. metadata-eval 48.8%

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

      9. associate-*l* 48.8%

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

      10. *-commutative 48.8%

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

      11. associate-/r* 48.8%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in i around inf 99.4%

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

      1. unpow2 99.4%

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

      2. associate-*r/ 99.4%

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

      3. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 99.3%

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

Alternative 2: 99.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{4 - \frac{1}{i \cdot i}} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (/ 0.25 (- 4.0 (/ 1.0 (* i i)))))

C

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 = 0.25d0 / (4.0d0 - (1.0d0 / (i * i)))
end function

Java

public static double code(double i) {
	return 0.25 / (4.0 - (1.0 / (i * i)));
}

Python

def code(i):
	return 0.25 / (4.0 - (1.0 / (i * i)))

Julia

function code(i)
	return Float64(0.25 / Float64(4.0 - Float64(1.0 / Float64(i * i))))
end

MATLAB

function tmp = code(i)
	tmp = 0.25 / (4.0 - (1.0 / (i * i)));
end

Wolfram

code[i_] := N[(0.25 / N[(4.0 - N[(1.0 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.4%

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

   1. times-frac 74.5%

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

   2. associate-/l* 74.4%

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

   3. associate-/l* 74.5%

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

   4. associate-/l/ 74.5%

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

   5. associate-/r/ 73.8%

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

   6. associate-/l* 73.8%

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

   7. *-inverses 73.8%

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

   8. metadata-eval 73.8%

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

   9. associate-*l* 73.8%

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

   10. *-commutative 73.8%

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

   11. associate-/r* 73.8%

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

3. Simplified 93.6%

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

4. Taylor expanded in i around 0 99.2%

   \[\leadsto \frac{0.25}{4 - \color{blue}{\frac{1}{{i}^{2}}}} \]
5. Step-by-step derivation

   1. unpow2 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 3: 98.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i * -0.25);
	} 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 = i * (i * (-0.25d0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 24.9%

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

      1. times-frac 99.9%

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

      2. associate-/l* 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-/l/ 99.7%

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

      5. associate-/r/ 98.4%

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

      6. associate-/l* 98.4%

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

      7. *-inverses 98.4%

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

      8. metadata-eval 98.4%

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

      9. associate-*l* 98.4%

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

      10. *-commutative 98.4%

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

      11. associate-/r* 98.5%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in i around 0 99.3%

      \[\leadsto \color{blue}{-0.25 \cdot {i}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l* 99.3%

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

   6. Simplified 99.3%

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

   if 0.5 < i

   1. Initial program 25.9%

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

      1. times-frac 48.8%

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

      2. associate-/l* 48.7%

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

      3. associate-/l* 48.8%

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

      4. associate-/l/ 48.8%

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

      5. associate-/r/ 48.8%

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

      6. associate-/l* 48.8%

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

      7. *-inverses 48.8%

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

      8. metadata-eval 48.8%

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

      9. associate-*l* 48.8%

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

      10. *-commutative 48.8%

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

      11. associate-/r* 48.8%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in i around inf 98.9%

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

4. Final simplification 99.1%

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

Alternative 4: 50.6% 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 25.4%

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

   1. times-frac 74.5%

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

   2. associate-/l* 74.4%

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

   3. associate-/l* 74.5%

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

   4. associate-/l/ 74.5%

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

   5. associate-/r/ 73.8%

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

   6. associate-/l* 73.8%

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

   7. *-inverses 73.8%

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

   8. metadata-eval 73.8%

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

   9. associate-*l* 73.8%

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

   10. *-commutative 73.8%

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

   11. associate-/r* 73.8%

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

3. Simplified 93.6%

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

4. Taylor expanded in i around inf 50.5%

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

5. Final simplification 50.5%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023188 
(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)))