Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 9.6s

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: 56.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4100000:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{6}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4100000.0)
   (* x (+ 1.0 (* (* y y) -0.16666666666666666)))
   (* x (/ 6.0 (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4100000.0) {
		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 <= 4100000.0d0) 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 <= 4100000.0) {
		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 <= 4100000.0:
		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 <= 4100000.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	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 <= 4100000.0)
		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, 4100000.0], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(6.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.1e6

   1. Initial program 99.9%

      \[x \cdot \frac{\sin y}{y} \]
   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. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 69.7%

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

   5. Simplified 69.7%

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

   if 4.1e6 < y

   1. Initial program 99.5%

      \[x \cdot \frac{\sin y}{y} \]
   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. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 2.1%

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

   5. Simplified 2.1%

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

      1. flip-+ N/A

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

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

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

   7. Applied egg-rr 1.1%

      \[\leadsto x \cdot \color{blue}{\frac{1 + \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot -0.027777777777777776}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}} \]

   8. Taylor expanded in y around 0

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

   10. Simplified 28.4%

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

   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 28.4%

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

   13. Simplified 28.4%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4100000:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{6}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.9% 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 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 70.1%

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

   if 2.39999999999999991 < y

   1. Initial program 99.5%

      \[x \cdot \frac{\sin y}{y} \]
   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. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 2.3%

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

   5. Simplified 2.3%

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

      1. flip-+ N/A

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

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

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

   7. Applied egg-rr 1.2%

      \[\leadsto x \cdot \color{blue}{\frac{1 + \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot -0.027777777777777776}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}} \]

   8. Taylor expanded in y around 0

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

   10. Simplified 27.9%

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

   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 27.9%

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

   13. Simplified 27.9%

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

Alternative 4: 63.0% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(1.0 / N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $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. *-lowering-*.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 55.5%

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

5. Simplified 55.5%

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

   1. flip-+ N/A

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

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

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

7. Applied egg-rr 55.0%

   \[\leadsto x \cdot \color{blue}{\frac{1 + \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot -0.027777777777777776}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}} \]

8. Taylor expanded in y around 0

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

10. Simplified 68.2%

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

Alternative 5: 63.0% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x / N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $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. *-lowering-*.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 55.5%

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

5. Simplified 55.5%

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

   1. flip-+ N/A

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

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

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

7. Applied egg-rr 55.0%

   \[\leadsto x \cdot \color{blue}{\frac{1 + \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot -0.027777777777777776}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}} \]

8. Taylor expanded in y around 0

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

10. Simplified 68.2%

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

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 68.2%

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

13. Simplified 68.2%

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

14. Final simplification 68.2%

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

Alternative 6: 56.4% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+84}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 7e+84) x 0.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e+84) {
		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 <= 7d+84) 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 <= 7e+84) {
		tmp = x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e+84:
		tmp = x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e+84)
		tmp = x;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e+84)
		tmp = x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e+84], x, 0.0]

Derivation

1. Split input into 2 regimes

2. if y < 6.9999999999999998e84

   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 66.2%

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

   if 6.9999999999999998e84 < y

   1. Initial program 99.5%

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

   4. Applied egg-rr 34.6%

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

Alternative 7: 15.7% 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 15.8%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))