Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.9s

Alternatives: 10

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 99.8%

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

   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 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: 63.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2:\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.2)
   (* (/ 6.0 y) (/ x y))
   (if (<= y 3e+28)
     (* 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 = (6.0 / y) * (x / y);
	} else if (y <= 3e+28) {
		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 = (6.0d0 / y) * (x / y)
    else if (y <= 3d+28) 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 = (6.0 / y) * (x / y);
	} else if (y <= 3e+28) {
		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 = (6.0 / y) * (x / y)
	elif y <= 3e+28:
		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(Float64(6.0 / y) * Float64(x / y));
	elseif (y <= 3e+28)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(x * 6.0) / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.2)
		tmp = (6.0 / y) * (x / y);
	elseif (y <= 3e+28)
		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[(N[(6.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+28], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.20000000000000018

   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 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 30.4%

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

      1. unpow2 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in y around inf 30.4%

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

      1. associate-*r/ 30.4%

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

      2. unpow2 30.4%

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

      3. times-frac 30.5%

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

   9. Simplified 30.5%

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

   if -6.20000000000000018 < y < 3.0000000000000001e28

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.4%

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

      1. unpow2 96.4%

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

   4. Simplified 96.4%

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

   if 3.0000000000000001e28 < y

   1. Initial program 99.6%

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

      1. clear-num 99.5%

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

      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 29.2%

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

      1. unpow2 29.2%

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

   6. Simplified 29.2%

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

   7. Taylor expanded in y around inf 29.2%

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

      1. associate-*r/ 29.2%

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

      2. unpow2 29.2%

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

   9. Simplified 29.2%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2:\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \]

Alternative 4: 63.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2:\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+28}:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.2)
   (* (/ 6.0 y) (/ x y))
   (if (<= y 2.1e+28)
     (+ 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 = (6.0 / y) * (x / y);
	} else if (y <= 2.1e+28) {
		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 = (6.0d0 / y) * (x / y)
    else if (y <= 2.1d+28) 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 = (6.0 / y) * (x / y);
	} else if (y <= 2.1e+28) {
		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 = (6.0 / y) * (x / y)
	elif y <= 2.1e+28:
		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(Float64(6.0 / y) * Float64(x / y));
	elseif (y <= 2.1e+28)
		tmp = Float64(x + Float64(-0.16666666666666666 * Float64(x * Float64(y * y))));
	else
		tmp = Float64(Float64(x * 6.0) / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.2)
		tmp = (6.0 / y) * (x / y);
	elseif (y <= 2.1e+28)
		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[(N[(6.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+28], N[(x + N[(-0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.20000000000000018

   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 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 30.4%

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

      1. unpow2 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in y around inf 30.4%

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

      1. associate-*r/ 30.4%

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

      2. unpow2 30.4%

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

      3. times-frac 30.5%

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

   9. Simplified 30.5%

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

   if -6.20000000000000018 < y < 2.09999999999999989e28

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \frac{\sin y}{y}} \cdot \sqrt[3]{x \cdot \frac{\sin y}{y}}\right) \cdot \sqrt[3]{x \cdot \frac{\sin y}{y}}} \]

      2. pow3 98.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \frac{\sin y}{y}}\right)}^{3}} \]

      3. *-commutative 98.1%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\sin y}{y} \cdot x}}\right)}^{3} \]

   3. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\sin y}{y} \cdot x}\right)}^{3}} \]

   4. Taylor expanded in y around 0 96.4%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot x + {y}^{2} \cdot \left(-0.1111111111111111 \cdot \left({1}^{0.3333333333333333} \cdot x\right) + -0.05555555555555555 \cdot \left({1}^{0.3333333333333333} \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 96.4%

         \[\leadsto \color{blue}{1} \cdot x + {y}^{2} \cdot \left(-0.1111111111111111 \cdot \left({1}^{0.3333333333333333} \cdot x\right) + -0.05555555555555555 \cdot \left({1}^{0.3333333333333333} \cdot x\right)\right) \]

      2. *-lft-identity 96.4%

         \[\leadsto \color{blue}{x} + {y}^{2} \cdot \left(-0.1111111111111111 \cdot \left({1}^{0.3333333333333333} \cdot x\right) + -0.05555555555555555 \cdot \left({1}^{0.3333333333333333} \cdot x\right)\right) \]

      3. unpow2 96.4%

         \[\leadsto x + \color{blue}{\left(y \cdot y\right)} \cdot \left(-0.1111111111111111 \cdot \left({1}^{0.3333333333333333} \cdot x\right) + -0.05555555555555555 \cdot \left({1}^{0.3333333333333333} \cdot x\right)\right) \]

      4. distribute-rgt-out 96.4%

         \[\leadsto x + \left(y \cdot y\right) \cdot \color{blue}{\left(\left({1}^{0.3333333333333333} \cdot x\right) \cdot \left(-0.1111111111111111 + -0.05555555555555555\right)\right)} \]

      5. pow-base-1 96.4%

         \[\leadsto x + \left(y \cdot y\right) \cdot \left(\left(\color{blue}{1} \cdot x\right) \cdot \left(-0.1111111111111111 + -0.05555555555555555\right)\right) \]

      6. *-lft-identity 96.4%

         \[\leadsto x + \left(y \cdot y\right) \cdot \left(\color{blue}{x} \cdot \left(-0.1111111111111111 + -0.05555555555555555\right)\right) \]

      7. metadata-eval 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in y around 0 96.4%

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

      1. unpow2 96.4%

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

      2. *-commutative 96.4%

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

   9. Simplified 96.4%

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

   if 2.09999999999999989e28 < y

   1. Initial program 99.6%

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

      1. clear-num 99.5%

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

      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 29.2%

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

      1. unpow2 29.2%

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

   6. Simplified 29.2%

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

   7. Taylor expanded in y around inf 29.2%

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

      1. associate-*r/ 29.2%

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

      2. unpow2 29.2%

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

   9. Simplified 29.2%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2:\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+28}:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \]

Alternative 5: 63.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.4\right):\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.5) (not (<= y 2.4))) (* (/ 6.0 y) (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.4)) {
		tmp = (6.0 / y) * (x / y);
	} else {
		tmp = 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 <= (-2.5d0)) .or. (.not. (y <= 2.4d0))) then
        tmp = (6.0d0 / y) * (x / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.4)) {
		tmp = (6.0 / y) * (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.5) or not (y <= 2.4):
		tmp = (6.0 / y) * (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.5) || !(y <= 2.4))
		tmp = Float64(Float64(6.0 / y) * Float64(x / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.5) || ~((y <= 2.4)))
		tmp = (6.0 / y) * (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.5], N[Not[LessEqual[y, 2.4]], $MachinePrecision]], N[(N[(6.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.5 or 2.39999999999999991 < 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 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 29.4%

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

      1. unpow2 29.4%

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

   6. Simplified 29.4%

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

   7. Taylor expanded in y around inf 29.4%

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

      1. associate-*r/ 29.4%

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

      2. unpow2 29.4%

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

      3. times-frac 29.4%

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

   9. Simplified 29.4%

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

   if -2.5 < y < 2.39999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.7%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.4\right):\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 63.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5)
   (* (/ 6.0 y) (/ x y))
   (if (<= y 2.4) x (/ (* x 6.0) (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5) {
		tmp = (6.0 / y) * (x / y);
	} else if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = (x * 6.0) / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.5d0)) then
        tmp = (6.0d0 / y) * (x / y)
    else if (y <= 2.4d0) then
        tmp = x
    else
        tmp = (x * 6.0d0) / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5) {
		tmp = (6.0 / y) * (x / y);
	} else if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = (x * 6.0) / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5:
		tmp = (6.0 / y) * (x / y)
	elif y <= 2.4:
		tmp = x
	else:
		tmp = (x * 6.0) / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5)
		tmp = Float64(Float64(6.0 / y) * Float64(x / y));
	elseif (y <= 2.4)
		tmp = x;
	else
		tmp = Float64(Float64(x * 6.0) / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5)
		tmp = (6.0 / y) * (x / y);
	elseif (y <= 2.4)
		tmp = x;
	else
		tmp = (x * 6.0) / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5], N[(N[(6.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4], x, N[(N[(x * 6.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5

   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 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 30.4%

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

      1. unpow2 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in y around inf 30.4%

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

      1. associate-*r/ 30.4%

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

      2. unpow2 30.4%

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

      3. times-frac 30.5%

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

   9. Simplified 30.5%

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

   if -2.5 < y < 2.39999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.7%

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

   if 2.39999999999999991 < y

   1. Initial program 99.6%

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

      1. clear-num 99.5%

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

      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 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. associate-*r/ 28.1%

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

      2. unpow2 28.1%

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

   9. Simplified 28.1%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \]

Alternative 7: 61.9% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+89} \lor \neg \left(y \leq 1.2 \cdot 10^{-23}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.2e+89) (not (<= y 1.2e-23))) (* y (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.2e+89) || !(y <= 1.2e-23)) {
		tmp = y * (x / y);
	} else {
		tmp = 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 <= (-2.2d+89)) .or. (.not. (y <= 1.2d-23))) then
        tmp = y * (x / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.2e+89) || !(y <= 1.2e-23)) {
		tmp = y * (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.2e+89) or not (y <= 1.2e-23):
		tmp = y * (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.2e+89) || !(y <= 1.2e-23))
		tmp = Float64(y * Float64(x / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.2e+89) || ~((y <= 1.2e-23)))
		tmp = y * (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.2e+89], N[Not[LessEqual[y, 1.2e-23]], $MachinePrecision]], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.2e89 or 1.19999999999999998e-23 < 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. clear-num 98.9%

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

      3. *-commutative 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in y around 0 6.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   5. Step-by-step derivation

      1. clear-num 6.5%

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

      2. *-inverses 6.5%

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

      3. associate-/l* 6.3%

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

      4. *-commutative 6.3%

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

      5. clear-num 6.3%

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

      6. associate-/r/ 6.3%

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

      7. *-commutative 6.3%

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

      8. associate-*r* 33.8%

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

      9. associate-/r/ 34.8%

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

      10. clear-num 33.8%

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

   6. Applied egg-rr 33.8%

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

   if -2.2e89 < y < 1.19999999999999998e-23

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 82.8%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+89} \lor \neg \left(y \leq 1.2 \cdot 10^{-23}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 62.2% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e+89) (/ y (/ y x)) (if (<= y 1.2e-23) x (* y (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+89) {
		tmp = y / (y / x);
	} else if (y <= 1.2e-23) {
		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 <= (-2.2d+89)) then
        tmp = y / (y / x)
    else if (y <= 1.2d-23) 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 <= -2.2e+89) {
		tmp = y / (y / x);
	} else if (y <= 1.2e-23) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e+89:
		tmp = y / (y / x)
	elif y <= 1.2e-23:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e+89)
		tmp = Float64(y / Float64(y / x));
	elseif (y <= 1.2e-23)
		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 <= -2.2e+89)
		tmp = y / (y / x);
	elseif (y <= 1.2e-23)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e+89], N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e-23], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e89

   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. clear-num 98.0%

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

      3. *-commutative 98.0%

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

   3. Applied egg-rr 98.0%

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

   4. Taylor expanded in y around 0 4.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   5. Step-by-step derivation

      1. clear-num 4.2%

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

      2. *-inverses 4.2%

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

      3. associate-/l* 4.0%

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

      4. *-commutative 4.0%

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

      5. clear-num 4.0%

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

      6. associate-/r/ 4.0%

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

      7. *-commutative 4.0%

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

      8. associate-*r* 38.7%

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

      9. associate-/r/ 41.0%

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

      10. clear-num 38.7%

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

   6. Applied egg-rr 38.7%

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

      1. *-commutative 38.7%

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

      2. clear-num 41.0%

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

      3. un-div-inv 41.0%

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

   8. Applied egg-rr 41.0%

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

   if -2.2e89 < y < 1.19999999999999998e-23

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 82.8%

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

   if 1.19999999999999998e-23 < y

   1. Initial program 99.5%

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

      1. associate-*r/ 99.4%

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

      2. clear-num 99.5%

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

      3. *-commutative 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around 0 8.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   5. Step-by-step derivation

      1. clear-num 8.3%

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

      2. *-inverses 8.3%

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

      3. associate-/l* 8.2%

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

      4. *-commutative 8.2%

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

      5. clear-num 8.2%

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

      6. associate-/r/ 8.2%

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

      7. *-commutative 8.2%

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

      8. associate-*r* 29.8%

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

      9. associate-/r/ 29.8%

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

      10. clear-num 29.8%

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

   6. Applied egg-rr 29.8%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 9: 63.7% 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 99.8%

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

   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 63.1%

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

   1. unpow2 63.1%

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

6. Simplified 63.1%

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

7. Final simplification 63.1%

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

Alternative 10: 51.8% 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 50.0%

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

3. Final simplification 50.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023176 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))