Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.9s

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. Final simplification 99.8%

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

Alternative 2: 57.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4) x (/ x (* y (* y 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = x / (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d0) then
        tmp = x
    else
        tmp = x / (y * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = x / (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4:
		tmp = x
	else:
		tmp = x / (y * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4)
		tmp = x;
	else
		tmp = Float64(x / Float64(y * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x;
	else
		tmp = x / (y * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(x / N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 67.2%

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

   if 2.39999999999999991 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.7%

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

      2. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in y around 0 19.3%

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

   8. Taylor expanded in y around inf 19.3%

      \[\leadsto \frac{x}{\color{blue}{\left(0.16666666666666666 \cdot y\right)} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.5% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 300000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 300000000.0) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 300000000.0) {
		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 <= 300000000.0d0) 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 <= 300000000.0) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 300000000.0:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 300000000.0)
		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 <= 300000000.0)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 300000000.0], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.9%

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

   if 3e8 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in y around 0 4.0%

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

      1. *-commutative 4.0%

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

   6. Simplified 4.0%

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

      1. *-un-lft-identity 4.0%

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

      2. times-frac 17.8%

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

      3. /-rgt-identity 17.8%

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

   8. Applied egg-rr 17.8%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 300000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.9% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2e-7) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-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 <= 2d-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 <= 2e-7) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-7:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-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 <= 2e-7)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-7], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 67.2%

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

   if 1.9999999999999999e-7 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.7%

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

   4. Taylor expanded in y around 0 3.7%

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

      1. *-commutative 3.7%

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

   6. Simplified 3.7%

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

      1. *-un-lft-identity 3.7%

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

      2. times-frac 16.5%

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

      3. /-rgt-identity 16.5%

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

   8. Applied egg-rr 16.5%

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

      1. clear-num 17.8%

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

      2. un-div-inv 17.8%

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

   10. Applied egg-rr 17.8%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.5% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot \left(y \cdot 0.16666666666666666\right) + 1} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ x (+ (* y (* y 0.16666666666666666)) 1.0)))

C

double code(double x, double y) {
	return x / ((y * (y * 0.16666666666666666)) + 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / ((y * (y * 0.16666666666666666d0)) + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return x / ((y * (y * 0.16666666666666666)) + 1.0);
}

Python

def code(x, y):
	return x / ((y * (y * 0.16666666666666666)) + 1.0)

Julia

function code(x, y)
	return Float64(x / Float64(Float64(y * Float64(y * 0.16666666666666666)) + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x / ((y * (y * 0.16666666666666666)) + 1.0);
end

Wolfram

code[x_, y_] := N[(x / N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*r/ 89.7%

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

   2. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.8%

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

   2. associate-/r/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in y around 0 63.4%

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

   1. *-commutative 63.4%

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

   2. distribute-rgt-in 63.4%

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

   3. *-commutative 63.4%

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

   4. lft-mult-inverse 63.5%

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

9. Applied egg-rr 63.5%

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

10. Final simplification 63.5%

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

Alternative 6: 51.5% 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 54.6%

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

4. Final simplification 54.6%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))