Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 10.6s

Alternatives: 9

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. Final simplification 99.8%

   \[\leadsto x \cdot \frac{\sin y}{y} \]

Alternative 2: 56.4% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.2)
   (* x (+ 1.0 (* (* y y) -0.16666666666666666)))
   (/ 6.0 (* y (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.2) {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 / (y * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.2d0) then
        tmp = x * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = 6.0d0 / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.2) {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.2:
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = 6.0 / (y * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.2)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(6.0 / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.2)
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = 6.0 / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.2], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.20000000000000018

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 67.9%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 67.9%

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

   4. Simplified 67.9%

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

   if 6.20000000000000018 < y

   1. Initial program 99.6%

      \[x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.5%

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

      2. associate-/l* 99.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

   4. Taylor expanded in y around 0 22.0%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 22.0%

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

   6. Simplified 22.0%

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

   7. Taylor expanded in y around inf 22.0%

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

      1. unpow2 22.0%

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

   9. Simplified 22.0%

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

       1. associate-*r/ 22.0%

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

       2. *-commutative 22.0%

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

       3. associate-/r* 22.1%

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

       4. *-commutative 22.1%

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

       5. associate-/l* 22.1%

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

       6. associate-/l/ 22.1%

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

   11. Applied egg-rr 22.1%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 3: 56.7% accurate, 11.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = 6.0 * (x / (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 = 6.0d0 * (x / (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 = 6.0 * (x / (y * y));
	}
	return tmp;
}

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4)
		tmp = x;
	else
		tmp = Float64(6.0 * Float64(x / 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 = 6.0 * (x / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(6.0 * N[(x / 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. Taylor expanded in y around 0 68.5%

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

   if 2.39999999999999991 < y

   1. Initial program 99.6%

      \[x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.5%

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

      2. associate-/l* 99.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

   4. Taylor expanded in y around 0 22.0%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 22.0%

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

   6. Simplified 22.0%

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

   7. Taylor expanded in y around inf 22.0%

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

      1. unpow2 22.0%

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

   9. Simplified 22.0%

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

4. Final simplification 56.9%

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

Alternative 4: 56.7% accurate, 11.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = 6.0 * ((x / 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 = 6.0d0 * ((x / 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 = 6.0 * ((x / y) / y);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(6.0 * N[(N[(x / y), $MachinePrecision] / y), $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. Taylor expanded in y around 0 68.5%

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

   if 2.39999999999999991 < y

   1. Initial program 99.6%

      \[x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.5%

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

      2. associate-/l* 99.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

   4. Taylor expanded in y around 0 22.0%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 22.0%

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

   6. Simplified 22.0%

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

   7. Taylor expanded in y around inf 22.0%

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

      1. unpow2 22.0%

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

   9. Simplified 22.0%

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

       1. associate-/r* 22.1%

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

       2. div-inv 22.1%

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

   11. Applied egg-rr 22.1%

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

       1. un-div-inv 22.1%

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

   13. Applied egg-rr 22.1%

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

4. Final simplification 56.9%

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

Alternative 5: 56.7% accurate, 11.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = 6.0 / (y * (y / x));
	}
	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 = 6.0d0 / (y * (y / x))
    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 = 6.0 / (y * (y / x));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(6.0 / N[(y * N[(y / x), $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. Taylor expanded in y around 0 68.5%

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

   if 2.39999999999999991 < y

   1. Initial program 99.6%

      \[x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.5%

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

      2. associate-/l* 99.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

   4. Taylor expanded in y around 0 22.0%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 22.0%

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

   6. Simplified 22.0%

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

   7. Taylor expanded in y around inf 22.0%

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

      1. unpow2 22.0%

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

   9. Simplified 22.0%

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

       1. associate-*r/ 22.0%

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

       2. *-commutative 22.0%

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

       3. associate-/r* 22.1%

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

       4. *-commutative 22.1%

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

       5. associate-/l* 22.1%

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

       6. associate-/l/ 22.1%

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

   11. Applied egg-rr 22.1%

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

4. Final simplification 56.9%

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

Alternative 6: 62.9% 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. Step-by-step derivation

   1. associate-*r/ 89.3%

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

   2. associate-/l* 99.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

4. Taylor expanded in y around 0 63.0%

   \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 63.0%

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

6. Simplified 63.0%

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

7. Final simplification 63.0%

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

Alternative 7: 56.0% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-32}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.5e-32) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-32) {
		tmp = x;
	} else {
		tmp = y * (x / 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 <= 2.5d-32) then
        tmp = x
    else
        tmp = y * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-32) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.5e-32:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5e-32)
		tmp = x;
	else
		tmp = Float64(y * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.5e-32)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e-32], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5e-32

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 67.7%

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

   if 2.5e-32 < y

   1. Initial program 99.6%

      \[x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 12.6%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{y} \]
   5. Step-by-step derivation

      1. *-commutative 12.6%

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

   6. Simplified 12.6%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{y} \]
   7. Step-by-step derivation

      1. associate-*r/ 24.7%

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

      2. *-commutative 24.7%

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

   8. Applied egg-rr 24.7%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-32}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 8: 56.4% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4e+16) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e+16) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d+16) then
        tmp = x
    else
        tmp = y / (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4e+16) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e+16:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e+16)
		tmp = x;
	else
		tmp = Float64(y / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e+16)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e+16], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4e16

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 68.0%

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

   if 4e16 < y

   1. Initial program 99.6%

      \[x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 3.8%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{y} \]
   5. Step-by-step derivation

      1. *-commutative 3.8%

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

   6. Simplified 3.8%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{y} \]
   7. Step-by-step derivation

      1. associate-*r/ 17.7%

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

      2. *-commutative 17.7%

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

   8. Applied egg-rr 17.7%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot y} \]
   9. Step-by-step derivation

      1. *-commutative 17.7%

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

      2. clear-num 21.2%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{y}{x}}} \]

      3. un-div-inv 21.2%

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

   10. Applied egg-rr 21.2%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]

Alternative 9: 51.2% accurate, 105.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.8%

   \[x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in y around 0 52.5%

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

3. Final simplification 52.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))