Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 63.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 42000000:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.4e+28)
   (/ (/ x y) (* y 0.16666666666666666))
   (if (<= y 42000000.0)
     (* x (+ 1.0 (* (* y y) -0.16666666666666666)))
     (/ (* x 6.0) (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.4e+28) {
		tmp = (x / y) / (y * 0.16666666666666666);
	} else if (y <= 42000000.0) {
		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 <= (-6.4d+28)) then
        tmp = (x / y) / (y * 0.16666666666666666d0)
    else if (y <= 42000000.0d0) 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 <= -6.4e+28) {
		tmp = (x / y) / (y * 0.16666666666666666);
	} else if (y <= 42000000.0) {
		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 <= -6.4e+28:
		tmp = (x / y) / (y * 0.16666666666666666)
	elif y <= 42000000.0:
		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 <= -6.4e+28)
		tmp = Float64(Float64(x / y) / Float64(y * 0.16666666666666666));
	elseif (y <= 42000000.0)
		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 <= -6.4e+28)
		tmp = (x / y) / (y * 0.16666666666666666);
	elseif (y <= 42000000.0)
		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, -6.4e+28], N[(N[(x / y), $MachinePrecision] / N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 42000000.0], 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 < -6.4000000000000001e28

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.5%

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

      1. associate-*r/ 99.6%

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

   4. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.5%

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

      3. div-inv 99.5%

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

      4. associate-/r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 31.8%

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

   8. Taylor expanded in y around inf 31.8%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{0.16666666666666666 \cdot y}} \]

   if -6.4000000000000001e28 < y < 4.2e7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 95.5%

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

      1. unpow2 95.5%

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

   4. Simplified 95.5%

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

   if 4.2e7 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. associate-*r/ 99.6%

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

   4. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.7%

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

      3. div-inv 99.5%

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

      4. associate-/r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 31.8%

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

   8. Taylor expanded in y around inf 31.8%

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

      1. associate-*r/ 31.8%

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

      2. unpow2 31.8%

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

   10. Simplified 31.8%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 42000000:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \]

Alternative 3: 63.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 42000000:\\ \;\;\;\;x + \left(y \cdot y\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.95e+35)
   (/ (/ x y) (* y 0.16666666666666666))
   (if (<= y 42000000.0)
     (+ x (* (* y y) (* x -0.16666666666666666)))
     (/ (* x 6.0) (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.95e+35) {
		tmp = (x / y) / (y * 0.16666666666666666);
	} else if (y <= 42000000.0) {
		tmp = x + ((y * y) * (x * -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 <= (-1.95d+35)) then
        tmp = (x / y) / (y * 0.16666666666666666d0)
    else if (y <= 42000000.0d0) then
        tmp = x + ((y * y) * (x * (-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 <= -1.95e+35) {
		tmp = (x / y) / (y * 0.16666666666666666);
	} else if (y <= 42000000.0) {
		tmp = x + ((y * y) * (x * -0.16666666666666666));
	} else {
		tmp = (x * 6.0) / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.95e+35:
		tmp = (x / y) / (y * 0.16666666666666666)
	elif y <= 42000000.0:
		tmp = x + ((y * y) * (x * -0.16666666666666666))
	else:
		tmp = (x * 6.0) / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.95e+35)
		tmp = Float64(Float64(x / y) / Float64(y * 0.16666666666666666));
	elseif (y <= 42000000.0)
		tmp = Float64(x + Float64(Float64(y * y) * Float64(x * -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 <= -1.95e+35)
		tmp = (x / y) / (y * 0.16666666666666666);
	elseif (y <= 42000000.0)
		tmp = x + ((y * y) * (x * -0.16666666666666666));
	else
		tmp = (x * 6.0) / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.95e+35], N[(N[(x / y), $MachinePrecision] / N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 42000000.0], N[(x + N[(N[(y * y), $MachinePrecision] * N[(x * -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 < -1.95e35

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.5%

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

      1. associate-*r/ 99.6%

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

   4. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.5%

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

      3. div-inv 99.5%

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

      4. associate-/r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 31.8%

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

   8. Taylor expanded in y around inf 31.8%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{0.16666666666666666 \cdot y}} \]

   if -1.95e35 < y < 4.2e7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 95.5%

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

      1. unpow2 95.5%

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

   4. Simplified 95.5%

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

      1. +-commutative 95.5%

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

      2. distribute-rgt-in 95.5%

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

      3. *-commutative 95.5%

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

      4. associate-*l* 95.5%

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

      5. *-un-lft-identity 95.5%

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

   6. Applied egg-rr 95.5%

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

   if 4.2e7 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. associate-*r/ 99.6%

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

   4. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.7%

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

      3. div-inv 99.5%

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

      4. associate-/r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 31.8%

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

   8. Taylor expanded in y around inf 31.8%

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

      1. associate-*r/ 31.8%

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

      2. unpow2 31.8%

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

   10. Simplified 31.8%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 42000000:\\ \;\;\;\;x + \left(y \cdot y\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \]

Alternative 4: 63.5% accurate, 9.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.5)) {
		tmp = (x * 6.0) / (y * 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 <= 2.5d0))) then
        tmp = (x * 6.0d0) / (y * 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 <= 2.5)) {
		tmp = (x * 6.0) / (y * y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -2.5 or 2.5 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.5%

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

      1. associate-*r/ 99.6%

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

   4. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.6%

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

      3. div-inv 99.4%

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

      4. associate-/r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 30.6%

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

   8. Taylor expanded in y around inf 30.6%

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

      1. associate-*r/ 30.6%

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

      2. unpow2 30.6%

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

   10. Simplified 30.6%

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

   if -2.5 < y < 2.5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.2%

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

4. Final simplification 66.5%

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

Alternative 5: 63.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5)
   (/ (/ x y) (* y 0.16666666666666666))
   (if (<= y 2.5) x (/ (* x 6.0) (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5) {
		tmp = (x / y) / (y * 0.16666666666666666);
	} else if (y <= 2.5) {
		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 = (x / y) / (y * 0.16666666666666666d0)
    else if (y <= 2.5d0) 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 = (x / y) / (y * 0.16666666666666666);
	} else if (y <= 2.5) {
		tmp = x;
	} else {
		tmp = (x * 6.0) / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5:
		tmp = (x / y) / (y * 0.16666666666666666)
	elif y <= 2.5:
		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(x / y) / Float64(y * 0.16666666666666666));
	elseif (y <= 2.5)
		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 = (x / y) / (y * 0.16666666666666666);
	elseif (y <= 2.5)
		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[(x / y), $MachinePrecision] / N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5], 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 99.8%

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

   2. Taylor expanded in x around 0 99.5%

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

      1. associate-*r/ 99.6%

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

   4. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.5%

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

      3. div-inv 99.4%

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

      4. associate-/r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 29.9%

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

   8. Taylor expanded in y around inf 29.9%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{0.16666666666666666 \cdot y}} \]

   if -2.5 < y < 2.5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.2%

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

   if 2.5 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. associate-*r/ 99.6%

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

   4. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.7%

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

      3. div-inv 99.4%

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

      4. associate-/r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 31.3%

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

   8. Taylor expanded in y around inf 31.3%

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

      1. associate-*r/ 31.3%

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

      2. unpow2 31.3%

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

   10. Simplified 31.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot y}\\ \end{array} \]

Alternative 6: 61.9% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+86} \lor \neg \left(y \leq 5 \cdot 10^{+23}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.3e+86) (not (<= y 5e+23))) (* y (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.3e+86) || !(y <= 5e+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 <= (-3.3d+86)) .or. (.not. (y <= 5d+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 <= -3.3e+86) || !(y <= 5e+23)) {
		tmp = y * (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.3e+86) or not (y <= 5e+23):
		tmp = y * (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.3e+86) || !(y <= 5e+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 <= -3.3e+86) || ~((y <= 5e+23)))
		tmp = y * (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.3e+86], N[Not[LessEqual[y, 5e+23]], $MachinePrecision]], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.2999999999999999e86 or 4.9999999999999999e23 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. associate-*r/ 99.6%

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

   4. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.7%

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

      3. div-inv 99.5%

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

      4. associate-/r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 34.8%

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

      1. associate-/r/ 34.8%

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

      2. clear-num 35.2%

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

      3. clear-num 35.2%

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

      4. clear-num 34.8%

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

   9. Applied egg-rr 34.8%

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

   if -3.2999999999999999e86 < y < 4.9999999999999999e23

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 85.1%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+86} \lor \neg \left(y \leq 5 \cdot 10^{+23}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 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 99.7%

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

   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 66.8%

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

   1. unpow2 66.8%

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

6. Simplified 66.8%

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

7. Final simplification 66.8%

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

Alternative 8: 51.6% 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 54.2%

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

3. Final simplification 54.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023174 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))