Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

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: 57.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.05e-7) x (* y (/ 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.05e-7) {
		tmp = x;
	} else {
		tmp = y * (1.0 / (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 <= 2.05d-7) then
        tmp = x
    else
        tmp = y * (1.0d0 / (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.05e-7) {
		tmp = x;
	} else {
		tmp = y * (1.0 / (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.05e-7:
		tmp = x
	else:
		tmp = y * (1.0 / (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.05e-7)
		tmp = x;
	else
		tmp = Float64(y * Float64(1.0 / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.05e-7)
		tmp = x;
	else
		tmp = y * (1.0 / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.05e-7], x, N[(y * N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.05e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 69.3%

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

   if 2.05e-7 < y

   1. Initial program 99.5%

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

      1. associate-*r/ 99.8%

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

      2. clear-num 99.1%

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

      3. associate-/r* 99.0%

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in y around 0 21.6%

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

      1. frac-2neg 21.6%

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

      2. associate-/r/ 21.6%

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

      3. distribute-neg-frac2 21.6%

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

   7. Applied egg-rr 21.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{-x}} \cdot \left(-y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.65e-6) x (/ 1.0 (/ (/ y x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-6) {
		tmp = x;
	} else {
		tmp = 1.0 / ((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 <= 1.65d-6) then
        tmp = x
    else
        tmp = 1.0d0 / ((y / x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-6) {
		tmp = x;
	} else {
		tmp = 1.0 / ((y / x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.65e-6:
		tmp = x
	else:
		tmp = 1.0 / ((y / x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.65e-6)
		tmp = x;
	else
		tmp = Float64(1.0 / Float64(Float64(y / x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.65e-6)
		tmp = x;
	else
		tmp = 1.0 / ((y / x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.65e-6], x, N[(1.0 / N[(N[(y / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.65000000000000008e-6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 69.3%

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

   if 1.65000000000000008e-6 < y

   1. Initial program 99.5%

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

      1. associate-*r/ 99.8%

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

      2. clear-num 99.1%

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

      3. associate-/r* 99.0%

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in y around 0 21.6%

      \[\leadsto \frac{1}{\frac{\frac{y}{x}}{\color{blue}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 57.9% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4e-7) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e-7) {
		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-7) 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-7) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e-7:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e-7)
		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-7)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e-7], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999998e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 69.3%

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

   if 3.9999999999999998e-7 < y

   1. Initial program 99.5%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x \cdot \sin y}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 5.5%

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

      1. *-commutative 5.5%

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

      2. associate-/l* 18.4%

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

   7. Applied egg-rr 18.4%

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

      1. clear-num 21.6%

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

      2. div-inv 21.6%

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

   9. Applied egg-rr 21.6%

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

Alternative 5: 57.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8e-29) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-29) {
		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 <= 8d-29) 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 <= 8e-29) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e-29:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e-29)
		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 <= 8e-29)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-29], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999955e-29

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 68.3%

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

   if 7.99999999999999955e-29 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x \cdot \sin y}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 16.5%

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

      1. *-commutative 16.5%

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

      2. associate-/l* 27.9%

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

   7. Applied egg-rr 27.9%

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

Alternative 6: 52.4% 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 56.4%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))