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. Final simplification 99.8%

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

Alternative 2: 58.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.14 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot 0.16666666666666666 + \frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.14e-10) x (/ (/ x y) (+ (* y 0.16666666666666666) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.14e-10) {
		tmp = x;
	} else {
		tmp = (x / y) / ((y * 0.16666666666666666) + (1.0 / 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 <= 1.14d-10) then
        tmp = x
    else
        tmp = (x / y) / ((y * 0.16666666666666666d0) + (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.14e-10) {
		tmp = x;
	} else {
		tmp = (x / y) / ((y * 0.16666666666666666) + (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.14e-10:
		tmp = x
	else:
		tmp = (x / y) / ((y * 0.16666666666666666) + (1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.14e-10)
		tmp = x;
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(y * 0.16666666666666666) + Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.14e-10)
		tmp = x;
	else
		tmp = (x / y) / ((y * 0.16666666666666666) + (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.14e-10], x, N[(N[(x / y), $MachinePrecision] / N[(N[(y * 0.16666666666666666), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1399999999999999e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.7%

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

   if 1.1399999999999999e-10 < y

   1. Initial program 99.7%

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

      1. clear-num 98.4%

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

      2. div-inv 98.4%

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

      3. div-inv 98.2%

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

      4. associate-/r* 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0 25.9%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{0.16666666666666666 \cdot y + \frac{1}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.14 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot 0.16666666666666666 + \frac{1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.6% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5000000000.0) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5000000000.0) {
		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 <= 5000000000.0d0) 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 <= 5000000000.0) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5000000000.0:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5000000000.0)
		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 <= 5000000000.0)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5000000000.0], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5e9

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.7%

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

   if 5e9 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 98.6%

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

      3. *-commutative 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in y around 0 4.6%

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

      1. clear-num 4.6%

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

      2. *-inverses 4.6%

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

      3. associate-/l* 4.4%

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

      4. *-commutative 4.4%

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

      5. div-inv 4.4%

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

      6. *-commutative 4.4%

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

      7. *-commutative 4.4%

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

      8. associate-*r* 23.1%

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

      9. associate-/r/ 23.4%

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

      10. clear-num 23.1%

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

   7. Applied egg-rr 23.1%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 10000.0) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 10000.0) {
		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 <= 10000.0d0) 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 <= 10000.0) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 10000.0:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 10000.0)
		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 <= 10000.0)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 10000.0], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.7%

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

   if 1e4 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 98.6%

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

      3. *-commutative 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in y around 0 4.6%

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

      1. clear-num 4.6%

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

      2. *-inverses 4.6%

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

      3. associate-/l* 4.4%

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

      4. *-commutative 4.4%

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

      5. div-inv 4.4%

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

      6. *-commutative 4.4%

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

      7. *-commutative 4.4%

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

      8. associate-*r* 22.5%

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

      9. associate-/r/ 22.8%

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

      10. clear-num 22.5%

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

   7. Applied egg-rr 22.5%

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

      1. *-commutative 22.5%

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

      2. clear-num 22.8%

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

      3. div-inv 22.8%

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

   9. Applied egg-rr 22.8%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]
5. 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 47.8%

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

4. Final simplification 47.8%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))