Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.7%

Time: 7.9s

Alternatives: 11

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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{\frac{y}{\sin y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ 1.0 (/ y (sin y)))))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return x * (1.0 / (y / Math.sin(y)));
}

Python

def code(x, y):
	return x * (1.0 / (y / math.sin(y)))

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(1.0 / N[(y / N[Sin[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 99.8%

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

   2. inv-pow 99.8%

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

3. Applied egg-rr 99.8%

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

   1. unpow-1 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 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 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. clear-num 99.8%

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

   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. Final simplification 99.8%

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

Alternative 4: 56.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 235000:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 235000.0)
   (+ x (* -0.16666666666666666 (* x (* y y))))
   (* (/ 6.0 y) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 235000.0) {
		tmp = x + (-0.16666666666666666 * (x * (y * y)));
	} else {
		tmp = (6.0 / 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 <= 235000.0d0) then
        tmp = x + ((-0.16666666666666666d0) * (x * (y * y)))
    else
        tmp = (6.0d0 / y) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 235000.0) {
		tmp = x + (-0.16666666666666666 * (x * (y * y)));
	} else {
		tmp = (6.0 / y) * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 235000.0:
		tmp = x + (-0.16666666666666666 * (x * (y * y)))
	else:
		tmp = (6.0 / y) * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 235000.0)
		tmp = Float64(x + Float64(-0.16666666666666666 * Float64(x * Float64(y * y))));
	else
		tmp = Float64(Float64(6.0 / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 235000.0)
		tmp = x + (-0.16666666666666666 * (x * (y * y)));
	else
		tmp = (6.0 / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 235000.0], N[(x + N[(-0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 235000

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 66.4%

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

      1. *-commutative 66.4%

         \[\leadsto x + -0.16666666666666666 \cdot \color{blue}{\left({y}^{2} \cdot x\right)} \]

      2. unpow2 66.4%

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

   4. Simplified 66.4%

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

   if 235000 < y

   1. Initial program 99.7%

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

      1. clear-num 99.8%

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

      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 29.0%

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

      1. *-commutative 29.0%

         \[\leadsto x \cdot \frac{1}{1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 29.0%

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

   6. Simplified 29.0%

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

   7. Taylor expanded in y around inf 29.0%

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

      1. associate-*r/ 29.0%

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

      2. unpow2 29.0%

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

      3. times-frac 29.1%

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

   9. Simplified 29.1%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 235000:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5: 57.0% accurate, 11.6× speedup

Math

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

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(6.0 / 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 = x * (6.0 / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.45], x, N[(x * N[(6.0 / 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 67.4%

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

   if 2.4500000000000002 < y

   1. Initial program 99.7%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 28.2%

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

      1. *-commutative 28.2%

         \[\leadsto x \cdot \frac{1}{1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 28.2%

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

   8. Simplified 28.2%

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

   9. Taylor expanded in y around inf 28.2%

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

       1. unpow2 28.2%

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

   11. Simplified 28.2%

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

4. Final simplification 57.4%

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

Alternative 6: 57.0% 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 67.4%

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

   if 2.4500000000000002 < y

   1. Initial program 99.7%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 28.2%

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

      1. *-commutative 28.2%

         \[\leadsto x \cdot \frac{1}{1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 28.2%

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

   8. Simplified 28.2%

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

   9. Taylor expanded in y around inf 28.2%

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

       1. unpow2 28.2%

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

       2. associate-/r* 28.2%

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

   11. Simplified 28.2%

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

4. Final simplification 57.4%

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

Alternative 7: 57.1% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.45)
		tmp = x;
	else
		tmp = Float64(Float64(6.0 / y) * Float64(x / 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 / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.45], x, N[(N[(6.0 / y), $MachinePrecision] * N[(x / 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 67.4%

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

   if 2.4500000000000002 < y

   1. Initial program 99.7%

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

      1. clear-num 99.8%

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

      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 28.2%

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

      1. *-commutative 28.2%

         \[\leadsto x \cdot \frac{1}{1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 28.2%

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

   6. Simplified 28.2%

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

   7. Taylor expanded in y around inf 28.2%

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

      1. associate-*r/ 28.2%

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

      2. unpow2 28.2%

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

      3. times-frac 28.2%

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

   9. Simplified 28.2%

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

4. Final simplification 57.5%

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

Alternative 8: 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. Step-by-step derivation

   1. clear-num 99.8%

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

   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 63.7%

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

   1. *-commutative 63.7%

      \[\leadsto x \cdot \frac{1}{1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}} \]

   2. unpow2 63.7%

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

6. Simplified 63.7%

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

7. Final simplification 63.7%

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

Alternative 9: 56.5% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.0) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.0) {
		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 <= 2.0d0) 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 <= 2.0) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.0:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.0)
		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 <= 2.0)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.0], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 67.4%

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

   if 2 < y

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 4.2%

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

      1. *-commutative 4.2%

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

   6. Simplified 4.2%

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

      1. *-un-lft-identity 4.2%

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

      2. times-frac 24.5%

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

      3. /-rgt-identity 24.5%

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

   8. Applied egg-rr 24.5%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 10: 56.8% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.8e-5) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-5) {
		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 <= 1.8d-5) 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 <= 1.8e-5) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e-5:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-5)
		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 <= 1.8e-5)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.8e-5], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000005e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 67.7%

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

   if 1.80000000000000005e-5 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.2%

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

   4. Simplified 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. div-inv 99.1%

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

      3. times-frac 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in y around 0 4.4%

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

      1. *-commutative 4.4%

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

   9. Simplified 4.4%

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

       1. associate-*l/ 4.4%

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

       2. *-un-lft-identity 4.4%

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

       3. associate-/l* 27.0%

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

   11. Applied egg-rr 27.0%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]

Alternative 11: 51.1% 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 51.4%

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

3. Final simplification 51.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))