Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

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: 56.6% accurate, 9.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.2) {
		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 <= 6.2d0) 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 <= 6.2) {
		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 <= 6.2:
		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 <= 6.2)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(x * Float64(Float64(6.0 / y) / y));
	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 = x * ((6.0 / y) / y);
	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[(x * N[(N[(6.0 / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.20000000000000018

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.1%

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

      1. unpow2 70.1%

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

   4. Simplified 70.1%

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

   if 6.20000000000000018 < y

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around 0 28.1%

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

      1. unpow2 28.1%

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

   6. Simplified 28.1%

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

   7. Taylor expanded in y around inf 28.1%

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

      1. *-commutative 28.1%

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

      2. unpow2 28.1%

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

      3. associate-*l/ 28.1%

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

   9. Simplified 28.1%

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

   10. Taylor expanded in x around 0 28.1%

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

       1. *-commutative 28.1%

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

       2. unpow2 28.1%

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

       3. associate-*l/ 28.1%

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

       4. associate-*r/ 28.1%

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

       5. associate-/r* 28.1%

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

   12. Simplified 28.1%

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

4. Final simplification 59.1%

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

Alternative 3: 56.6% accurate, 9.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.2) {
		tmp = x + (-0.16666666666666666 * (x * (y * y)));
	} 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 <= 6.2d0) then
        tmp = x + ((-0.16666666666666666d0) * (x * (y * y)))
    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 <= 6.2) {
		tmp = x + (-0.16666666666666666 * (x * (y * y)));
	} else {
		tmp = x * ((6.0 / y) / y);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 6.20000000000000018

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.2%

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

      1. unpow2 70.2%

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

   4. Simplified 70.2%

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

   if 6.20000000000000018 < y

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around 0 28.1%

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

      1. unpow2 28.1%

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

   6. Simplified 28.1%

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

   7. Taylor expanded in y around inf 28.1%

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

      1. *-commutative 28.1%

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

      2. unpow2 28.1%

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

      3. associate-*l/ 28.1%

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

   9. Simplified 28.1%

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

   10. Taylor expanded in x around 0 28.1%

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

       1. *-commutative 28.1%

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

       2. unpow2 28.1%

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

       3. associate-*l/ 28.1%

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

       4. associate-*r/ 28.1%

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

       5. associate-/r* 28.1%

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

   12. Simplified 28.1%

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

4. Final simplification 59.1%

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

Alternative 4: 56.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.45) x (* 6.0 (/ x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.45) {
		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 <= 2.45d0) 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 <= 2.45) {
		tmp = x;
	} else {
		tmp = 6.0 * (x / (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.45:
		tmp = x
	else:
		tmp = 6.0 * (x / (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.45)
		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 <= 2.45)
		tmp = x;
	else
		tmp = 6.0 * (x / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.45], x, N[(6.0 * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4500000000000002

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.8%

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

   if 2.4500000000000002 < y

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around 0 28.1%

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

      1. unpow2 28.1%

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

   6. Simplified 28.1%

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

   7. Taylor expanded in y around inf 28.1%

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

      1. unpow2 28.1%

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

   9. Simplified 28.1%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \end{array} \]

Alternative 5: 56.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{6}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.45) x (* x (/ (/ 6.0 y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.45) {
		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 <= 2.45d0) 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 <= 2.45) {
		tmp = x;
	} else {
		tmp = x * ((6.0 / y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.45:
		tmp = x
	else:
		tmp = x * ((6.0 / y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.45)
		tmp = x;
	else
		tmp = Float64(x * Float64(Float64(6.0 / y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.45)
		tmp = x;
	else
		tmp = x * ((6.0 / y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.45], x, N[(x * N[(N[(6.0 / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4500000000000002

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.8%

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

   if 2.4500000000000002 < y

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around 0 28.1%

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

      1. unpow2 28.1%

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

   6. Simplified 28.1%

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

   7. Taylor expanded in y around inf 28.1%

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

      1. *-commutative 28.1%

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

      2. unpow2 28.1%

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

      3. associate-*l/ 28.1%

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

   9. Simplified 28.1%

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

   10. Taylor expanded in x around 0 28.1%

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

       1. *-commutative 28.1%

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

       2. unpow2 28.1%

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

       3. associate-*l/ 28.1%

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

       4. associate-*r/ 28.1%

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

       5. associate-/r* 28.1%

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

   12. Simplified 28.1%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{6}{y}}{y}\\ \end{array} \]

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. clear-num 99.7%

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

   2. un-div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around 0 65.5%

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

   1. unpow2 65.5%

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

6. Simplified 65.5%

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

7. Final simplification 65.5%

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

Alternative 7: 56.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1e-19) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e-19) {
		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 <= 1d-19) 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 <= 1e-19) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e-19:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e-19)
		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 <= 1e-19)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e-19], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999998e-20

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.5%

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

   if 9.9999999999999998e-20 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 28.9%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 8: 56.5% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.08e-10) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.08e-10) {
		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 <= 1.08d-10) 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 <= 1.08e-10) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.08e-10:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.08e-10)
		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 <= 1.08e-10)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.08e-10], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.08000000000000002e-10

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.5%

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

   if 1.08000000000000002e-10 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 28.9%

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

      1. clear-num 28.9%

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

      2. un-div-inv 28.9%

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

   6. Applied egg-rr 28.9%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]

Alternative 9: 50.5% 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 53.4%

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

3. Final simplification 53.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))