Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.0% → 99.7%

Time: 4.6s

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: 27.0% 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: 99.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{i}{i \cdot 16 - \frac{4}{i}} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (/ i (- (* i 16.0) (/ 4.0 i))))

C

double code(double i) {
	return i / ((i * 16.0) - (4.0 / i));
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = i / ((i * 16.0d0) - (4.0d0 / i))
end function

Java

public static double code(double i) {
	return i / ((i * 16.0) - (4.0 / i));
}

Python

def code(i):
	return i / ((i * 16.0) - (4.0 / i))

Julia

function code(i)
	return Float64(i / Float64(Float64(i * 16.0) - Float64(4.0 / i)))
end

MATLAB

function tmp = code(i)
	tmp = i / ((i * 16.0) - (4.0 / i));
end

Wolfram

code[i_] := N[(i / N[(N[(i * 16.0), $MachinePrecision] - N[(4.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-/l/ 28.6%

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

   2. sqr-neg 28.6%

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

   3. associate-*l* 28.5%

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

   4. swap-sqr 28.5%

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

   5. sqr-neg 28.5%

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

   6. swap-sqr 28.5%

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

   7. associate-/l* 33.9%

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

   8. associate-/l* 36.5%

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

3. Simplified 72.6%

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

5. Taylor expanded in i around 0 99.9%

   \[\leadsto \frac{i}{\color{blue}{16 \cdot i - 4 \cdot \frac{1}{i}}} \]
6. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \frac{i}{\color{blue}{i \cdot 16} - 4 \cdot \frac{1}{i}} \]

   2. associate-*r/ 99.9%

      \[\leadsto \frac{i}{i \cdot 16 - \color{blue}{\frac{4 \cdot 1}{i}}} \]

   3. metadata-eval 99.9%

      \[\leadsto \frac{i}{i \cdot 16 - \frac{\color{blue}{4}}{i}} \]

7. Simplified 99.9%

   \[\leadsto \frac{i}{\color{blue}{i \cdot 16 - \frac{4}{i}}} \]

8. Final simplification 99.9%

   \[\leadsto \frac{i}{i \cdot 16 - \frac{4}{i}} \]
9. Add Preprocessing

Alternative 2: 98.9% accurate, 2.3× speedup

Math

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

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 * Float64(-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 32.4%

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

      1. associate-/l/ 32.4%

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

      2. sqr-neg 32.4%

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

      3. associate-*l* 32.3%

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

      4. swap-sqr 32.3%

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

      5. sqr-neg 32.3%

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

      6. swap-sqr 32.3%

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

      7. associate-/l* 43.1%

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

      8. associate-/l* 43.2%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in i around 0 98.6%

      \[\leadsto \frac{i}{\color{blue}{\frac{-4}{i}}} \]
   6. Step-by-step derivation

      1. frac-2neg 98.6%

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

      2. metadata-eval 98.6%

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

      3. associate-/r/ 98.9%

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

      4. div-inv 98.9%

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

      5. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

   if 0.5 < i

   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. associate-/l/ 24.8%

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

      2. sqr-neg 24.8%

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

      3. associate-*l* 24.7%

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

      4. swap-sqr 24.7%

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

      5. sqr-neg 24.7%

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

      6. swap-sqr 24.7%

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

      7. associate-/l* 24.9%

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

      8. associate-/l* 29.8%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in i around inf 99.0%

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

4. Final simplification 99.0%

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

Alternative 3: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\frac{i}{\frac{-4}{i}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (if (<= i 0.5) (/ i (/ -4.0 i)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i / (-4.0 / 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 = i / ((-4.0d0) / 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 = i / (-4.0 / i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = i / (-4.0 / i)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(i / Float64(-4.0 / i));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = i / (-4.0 / i);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(i / N[(-4.0 / i), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 32.4%

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

      1. associate-/l/ 32.4%

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

      2. sqr-neg 32.4%

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

      3. associate-*l* 32.3%

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

      4. swap-sqr 32.3%

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

      5. sqr-neg 32.3%

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

      6. swap-sqr 32.3%

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

      7. associate-/l* 43.1%

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

      8. associate-/l* 43.2%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in i around 0 98.6%

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

   if 0.5 < i

   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. associate-/l/ 24.8%

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

      2. sqr-neg 24.8%

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

      3. associate-*l* 24.7%

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

      4. swap-sqr 24.7%

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

      5. sqr-neg 24.7%

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

      6. swap-sqr 24.7%

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

      7. associate-/l* 24.9%

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

      8. associate-/l* 29.8%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in i around inf 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\frac{i}{\frac{-4}{i}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

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 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. Step-by-step derivation

   1. associate-/l/ 28.6%

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

   2. sqr-neg 28.6%

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

   3. associate-*l* 28.5%

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

   4. swap-sqr 28.5%

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

   5. sqr-neg 28.5%

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

   6. swap-sqr 28.5%

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

   7. associate-/l* 33.9%

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

   8. associate-/l* 36.5%

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

3. Simplified 72.6%

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

5. Taylor expanded in i around inf 51.2%

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

6. Final simplification 51.2%

   \[\leadsto 0.0625 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))