Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 9.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return x * (Math.sin(y) / y);
}

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (sin(y) / y);
end

Wolfram

code[x_, y_] := N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return x * (Math.sin(y) / y);
}

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (sin(y) / y);
end

Wolfram

code[x_, y_] := N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return x * (Math.sin(y) / y);
}

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (sin(y) / y);
end

Wolfram

code[x_, y_] := N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 58.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 6}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e+24)
   (* x (+ 1.0 (* y (* y -0.16666666666666666))))
   (/ (/ (* x 6.0) y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e+24) {
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = ((x * 6.0) / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.2d+24) then
        tmp = x * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
    else
        tmp = ((x * 6.0d0) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.2e+24) {
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = ((x * 6.0) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.2e+24:
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)))
	else:
		tmp = ((x * 6.0) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e+24)
		tmp = Float64(x * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))));
	else
		tmp = Float64(Float64(Float64(x * 6.0) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.2e+24)
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)));
	else
		tmp = ((x * 6.0) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e+24], N[(x * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 6.0), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e24

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

      3. associate-*l* N/A

         \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 61.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 61.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]

   if 1.2e24 < y

   1. Initial program 99.7%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

      3. associate-*l* N/A

         \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 2.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 2.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}}}\right)\right)\right) \]

      5. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{1}{1 + \color{blue}{y \cdot \left(y \cdot \frac{-1}{6}\right)}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 2.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 2.1%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{1}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 24.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 24.5%

       \[\leadsto x \cdot \frac{1}{\color{blue}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}} \]

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{6 \cdot x}{\color{blue}{{y}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{6 \cdot x}{y \cdot \color{blue}{y}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{6 \cdot x}{y}}{\color{blue}{y}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{6 \cdot x}{y}\right), \color{blue}{y}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(6 \cdot x\right), y\right), y\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot 6\right), y\right), y\right) \]

       7. *-lowering-*.f64 24.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), y\right), y\right) \]

   13. Simplified 24.5%

       \[\leadsto \color{blue}{\frac{\frac{x \cdot 6}{y}}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 58.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 6}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.4) x (/ (/ (* x 6.0) y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = ((x * 6.0) / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d0) then
        tmp = x
    else
        tmp = ((x * 6.0d0) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = ((x * 6.0) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4:
		tmp = x
	else:
		tmp = ((x * 6.0) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4)
		tmp = x;
	else
		tmp = Float64(Float64(Float64(x * 6.0) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x;
	else
		tmp = ((x * 6.0) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(N[(N[(x * 6.0), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 63.6%

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

   if 2.39999999999999991 < y

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

      3. associate-*l* N/A

         \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 2.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}}}\right)\right)\right) \]

      5. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{1}{1 + \color{blue}{y \cdot \left(y \cdot \frac{-1}{6}\right)}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 2.6%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{1}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 22.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 22.6%

       \[\leadsto x \cdot \frac{1}{\color{blue}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}} \]

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{6 \cdot x}{\color{blue}{{y}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{6 \cdot x}{y \cdot \color{blue}{y}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{6 \cdot x}{y}}{\color{blue}{y}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{6 \cdot x}{y}\right), \color{blue}{y}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(6 \cdot x\right), y\right), y\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot 6\right), y\right), y\right) \]

       7. *-lowering-*.f64 22.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), y\right), y\right) \]

   13. Simplified 22.6%

       \[\leadsto \color{blue}{\frac{\frac{x \cdot 6}{y}}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 58.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{6}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.4) x (* x (/ 6.0 (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = x * (6.0 / (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d0) then
        tmp = x
    else
        tmp = x * (6.0d0 / (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = x * (6.0 / (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4:
		tmp = x
	else:
		tmp = x * (6.0 / (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4)
		tmp = x;
	else
		tmp = Float64(x * Float64(6.0 / Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x;
	else
		tmp = x * (6.0 / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(x * N[(6.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 63.6%

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

   if 2.39999999999999991 < y

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

      3. associate-*l* N/A

         \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 2.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}}}\right)\right)\right) \]

      5. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{1}{1 + \color{blue}{y \cdot \left(y \cdot \frac{-1}{6}\right)}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 2.6%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{1}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 22.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 22.6%

       \[\leadsto x \cdot \frac{1}{\color{blue}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}} \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{6}{{y}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(6, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(6, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

       3. *-lowering-*.f64 22.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Simplified 22.6%

       \[\leadsto x \cdot \color{blue}{\frac{6}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 64.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ x (+ 1.0 (* 0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	return x / (1.0 + (0.16666666666666666 * (y * y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (1.0d0 + (0.16666666666666666d0 * (y * y)))
end function

Java

public static double code(double x, double y) {
	return x / (1.0 + (0.16666666666666666 * (y * y)));
}

Python

def code(x, y):
	return x / (1.0 + (0.16666666666666666 * (y * y)))

Julia

function code(x, y)
	return Float64(x / Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (1.0 + (0.16666666666666666 * (y * y)));
end

Wolfram

code[x_, y_] := N[(x / N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

   2. *-commutative N/A

      \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

   3. associate-*l* N/A

      \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

   4. *-commutative N/A

      \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

   5. distribute-lft-in N/A

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 47.7%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

5. Simplified 47.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}\right)}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) - 1 \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right)}}}\right)\right)\right) \]

   5. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{1}{1 + \color{blue}{y \cdot \left(y \cdot \frac{-1}{6}\right)}}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 47.7%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

7. Applied egg-rr 47.7%

   \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{1}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 60.0%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right) \]

10. Simplified 60.0%

    \[\leadsto x \cdot \frac{1}{\color{blue}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}} \]

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{x}{1 + \frac{1}{6} \cdot {y}^{2}}} \]
12. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

    5. *-lowering-*.f64 60.0%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

13. Simplified 60.0%

    \[\leadsto \color{blue}{\frac{x}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)}} \]
14. Add Preprocessing

Alternative 6: 58.2% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+42}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.7e+42) x 0.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+42) {
		tmp = x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d+42) then
        tmp = x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+42) {
		tmp = x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e+42:
		tmp = x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+42)
		tmp = x;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e+42)
		tmp = x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e+42], x, 0.0]

Derivation

1. Split input into 2 regimes

2. if y < 2.7000000000000001e42

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 61.0%

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

   if 2.7000000000000001e42 < y

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   4. Applied egg-rr 25.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 15.9% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 99.8%

   \[x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

4. Applied egg-rr 15.4%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))