Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.9s

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: 99.8% 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 99.7%

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

   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. Final simplification 99.8%

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

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: 58.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 110000:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{y}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 110000.0)
   (* x (+ 1.0 (* (* y y) -0.16666666666666666)))
   (/ 6.0 (/ y (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 110000.0) {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	} 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 <= 110000.0d0) then
        tmp = x * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    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 <= 110000.0) {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 / (y / (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 110000.0:
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = 6.0 / (y / (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 110000.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(6.0 / Float64(y / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 110000.0)
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = 6.0 / (y / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 110000.0], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e5

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 68.8%

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

      1. unpow2 68.8%

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

   4. Simplified 68.8%

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

   if 1.1e5 < y

   1. Initial program 99.4%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 27.8%

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

      1. unpow2 27.8%

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

   6. Simplified 27.8%

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

   7. Taylor expanded in y around inf 27.8%

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

      1. unpow2 27.8%

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

   9. Simplified 27.8%

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

       1. clear-num 27.8%

          \[\leadsto 6 \cdot \color{blue}{\frac{1}{\frac{y \cdot y}{x}}} \]

       2. un-div-inv 27.8%

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

       3. associate-/l* 27.8%

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

   11. Applied egg-rr 27.8%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 110000:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{y}{\frac{x}{y}}}\\ \end{array} \]

Alternative 4: 58.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 110000:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{y}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 110000.0)
   (* x (+ 1.0 (* y (* y -0.16666666666666666))))
   (/ 6.0 (/ y (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 110000.0) {
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)));
	} 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 <= 110000.0d0) then
        tmp = x * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
    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 <= 110000.0) {
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = 6.0 / (y / (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 110000.0:
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)))
	else:
		tmp = 6.0 / (y / (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 110000.0)
		tmp = Float64(x * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))));
	else
		tmp = Float64(6.0 / Float64(y / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 110000.0)
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)));
	else
		tmp = 6.0 / (y / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 110000.0], N[(x * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e5

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 68.8%

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

      1. unpow2 68.8%

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

   4. Simplified 68.8%

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

   5. Taylor expanded in y around 0 68.8%

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

      1. unpow2 68.8%

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

      2. *-commutative 68.8%

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

      3. associate-*r* 68.8%

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

   7. Simplified 68.8%

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

   if 1.1e5 < y

   1. Initial program 99.4%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 27.8%

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

      1. unpow2 27.8%

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

   6. Simplified 27.8%

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

   7. Taylor expanded in y around inf 27.8%

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

      1. unpow2 27.8%

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

   9. Simplified 27.8%

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

       1. clear-num 27.8%

          \[\leadsto 6 \cdot \color{blue}{\frac{1}{\frac{y \cdot y}{x}}} \]

       2. un-div-inv 27.8%

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

       3. associate-/l* 27.8%

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

   11. Applied egg-rr 27.8%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 110000:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{y}{\frac{x}{y}}}\\ \end{array} \]

Alternative 5: 58.2% 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 99.9%

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

   2. Taylor expanded in y around 0 69.9%

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

   if 2.39999999999999991 < y

   1. Initial program 99.4%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 26.6%

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

      1. unpow2 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in y around inf 26.6%

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

      1. unpow2 26.6%

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

   9. Simplified 26.6%

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

4. Final simplification 58.3%

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

Alternative 6: 58.2% 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 99.9%

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

   2. Taylor expanded in y around 0 69.9%

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

   if 2.39999999999999991 < y

   1. Initial program 99.4%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 26.6%

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

      1. unpow2 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in y around inf 26.6%

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

      1. unpow2 26.6%

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

      2. associate-/r* 26.7%

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

   9. Simplified 26.7%

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

4. Final simplification 58.3%

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

Alternative 7: 58.2% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(6.0 / N[(y / N[(x / 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. Taylor expanded in y around 0 69.9%

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

   if 2.39999999999999991 < y

   1. Initial program 99.4%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 26.6%

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

      1. unpow2 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in y around inf 26.6%

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

      1. unpow2 26.6%

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

   9. Simplified 26.6%

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

       1. clear-num 26.6%

          \[\leadsto 6 \cdot \color{blue}{\frac{1}{\frac{y \cdot y}{x}}} \]

       2. un-div-inv 26.6%

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

       3. associate-/l* 26.7%

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

   11. Applied egg-rr 26.7%

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

4. Final simplification 58.3%

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

Alternative 8: 64.1% 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 99.7%

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

   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 65.3%

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

   1. unpow2 65.3%

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

6. Simplified 65.3%

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

   1. add-sqr-sqrt 65.3%

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

   2. sqrt-unprod 65.3%

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

   3. swap-sqr 65.3%

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

   4. metadata-eval 65.3%

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

   5. metadata-eval 65.3%

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

   6. swap-sqr 65.3%

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

   7. sqrt-unprod 25.4%

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

   8. add-sqr-sqrt 64.9%

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

   9. add-log-exp 64.7%

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

   10. exp-prod 64.7%

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

8. Applied egg-rr 64.7%

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

   1. log-pow 64.9%

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

   2. rem-log-exp 64.9%

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

   3. unpow2 64.9%

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

   4. *-commutative 64.9%

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

   5. unpow2 64.9%

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

10. Simplified 64.9%

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

11. Final simplification 64.9%

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

Alternative 9: 64.4% 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 65.3%

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

   1. unpow2 65.3%

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

6. Simplified 65.3%

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

7. Final simplification 65.3%

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

Alternative 10: 57.4% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5e+72) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+72) {
		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 <= 5d+72) 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 <= 5e+72) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+72:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+72)
		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 <= 5e+72)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+72], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999992e72

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 64.9%

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

   if 4.99999999999999992e72 < y

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in y around 0 3.6%

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

      1. associate-/l* 30.4%

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

      2. div-inv 30.4%

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

      3. clear-num 28.7%

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

   5. Applied egg-rr 28.7%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 11: 52.0% 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 52.3%

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

3. Final simplification 52.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))