Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 6

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 57.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 98000:\\ \;\;\;\;x + \left(x \cdot -0.16666666666666666\right) \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 98000.0)
   (+ x (* (* x -0.16666666666666666) (* y y)))
   (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 98000.0) {
		tmp = x + ((x * -0.16666666666666666) * (y * y));
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 98000.0d0) then
        tmp = x + ((x * (-0.16666666666666666d0)) * (y * y))
    else
        tmp = y / (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 98000.0) {
		tmp = x + ((x * -0.16666666666666666) * (y * y));
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 98000.0:
		tmp = x + ((x * -0.16666666666666666) * (y * y))
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 98000.0)
		tmp = Float64(x + Float64(Float64(x * -0.16666666666666666) * Float64(y * y)));
	else
		tmp = Float64(y / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 98000.0)
		tmp = x + ((x * -0.16666666666666666) * (y * y));
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 98000.0], N[(x + N[(N[(x * -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 98000

   1. Initial program 99.9%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 67.6%

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

      1. associate-*r* 67.6%

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

      2. *-commutative 67.6%

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

   5. Simplified 67.6%

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

      1. unpow2 67.6%

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

   7. Applied egg-rr 67.6%

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

   if 98000 < y

   1. Initial program 99.7%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. clear-num 99.2%

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

      2. un-div-inv 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in y around 0 26.3%

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

Alternative 3: 57.1% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4e+20) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e+20) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d+20) then
        tmp = x
    else
        tmp = y / (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4e+20) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e+20:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e+20)
		tmp = x;
	else
		tmp = Float64(y / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e+20)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e+20], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4e20

   1. Initial program 99.9%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 66.1%

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

   if 4e20 < y

   1. Initial program 99.7%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. clear-num 99.2%

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

      2. un-div-inv 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in y around 0 27.5%

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

Alternative 4: 56.9% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2e+21) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e+21) {
		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 <= 2d+21) 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 <= 2e+21) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e+21:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e+21)
		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 <= 2e+21)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e+21], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e21

   1. Initial program 99.9%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 66.1%

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

   if 2e21 < y

   1. Initial program 99.7%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 26.4%

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

Alternative 5: 52.1% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.8%

   \[x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 49.8%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Alternative 6: 3.6% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.8%

   \[x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 86.6%

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

   1. associate-*l/ 88.8%

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

   2. *-commutative 88.8%

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

5. Simplified 88.8%

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

6. Taylor expanded in y around 0 49.4%

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

   1. clear-num 49.7%

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

   2. associate-/r/ 48.9%

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

8. Applied egg-rr 48.9%

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

10. Applied egg-rr 2.3%

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

    1. *-inverses 3.4%

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

12. Simplified 3.4%

    \[\leadsto \color{blue}{1} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))