Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.2s

Alternatives: 5

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: 56.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.2e-20) x (* y (/ 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-20) {
		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 <= 2.2d-20) 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 <= 2.2e-20) {
		tmp = x;
	} else {
		tmp = y * (1.0 / (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e-20:
		tmp = x
	else:
		tmp = y * (1.0 / (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-20)
		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 <= 2.2e-20)
		tmp = x;
	else
		tmp = y * (1.0 / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-20], x, N[(y * N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.19999999999999991e-20

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 65.3%

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

   if 2.19999999999999991e-20 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 7.0%

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

      1. *-commutative 7.0%

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

      2. associate-/l* 23.7%

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

   7. Applied egg-rr 23.7%

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

      1. *-commutative 23.7%

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

      2. div-inv 23.7%

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

      3. associate-*l* 7.1%

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

      4. lft-mult-inverse 7.1%

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

      5. metadata-eval 7.1%

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

      6. div-inv 7.1%

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

      7. clear-num 7.1%

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

   9. Applied egg-rr 7.1%

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

       1. *-inverses 7.1%

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

       2. associate-/r* 25.6%

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

       3. div-inv 25.6%

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

       4. frac-2neg 25.6%

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

       5. div-inv 25.6%

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

       6. distribute-neg-frac2 25.6%

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

   11. Applied egg-rr 25.6%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.8e-33) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.8e-33) {
		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 <= 5.8d-33) 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 <= 5.8e-33) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.8e-33:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.8e-33)
		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 <= 5.8e-33)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.8e-33], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.80000000000000005e-33

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 64.6%

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

   if 5.80000000000000005e-33 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in y around 0 11.7%

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

      1. *-commutative 11.7%

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

      2. associate-/l* 28.8%

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

   7. Applied egg-rr 28.8%

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

      1. clear-num 30.6%

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

      2. un-div-inv 30.6%

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

   9. Applied egg-rr 30.6%

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

Alternative 4: 56.1% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.005) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.005) {
		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 <= 0.005d0) 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 <= 0.005) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.005:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.005)
		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 <= 0.005)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.005], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0050000000000000001

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 65.4%

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

   if 0.0050000000000000001 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 4.5%

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

      1. *-commutative 4.5%

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

      2. associate-/l* 21.8%

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

   7. Applied egg-rr 21.8%

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

Alternative 5: 50.5% 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 52.6%

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

Reproduce

herbie shell --seed 2024086 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))