Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

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: 57.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4.4e-8) x (* y (/ 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-8) {
		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 <= 4.4d-8) 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 <= 4.4e-8) {
		tmp = x;
	} else {
		tmp = y * (1.0 / (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.4e-8:
		tmp = x
	else:
		tmp = y * (1.0 / (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e-8)
		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 <= 4.4e-8)
		tmp = x;
	else
		tmp = y * (1.0 / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e-8], x, N[(y * N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3999999999999997e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 71.4%

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

   if 4.3999999999999997e-8 < y

   1. Initial program 99.5%

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

      1. associate-*r/ 99.7%

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

      2. clear-num 98.4%

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

      3. *-commutative 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in y around 0 7.3%

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

      1. associate-/r/ 7.3%

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

      2. metadata-eval 7.3%

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

      3. *-inverses 7.3%

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

      4. associate-/r/ 27.1%

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

      5. div-inv 27.1%

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

   7. Applied egg-rr 27.1%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.6e-8) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-8) {
		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 <= 2.6d-8) 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 <= 2.6e-8) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e-8:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e-8)
		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 <= 2.6e-8)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e-8], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6000000000000001e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 71.4%

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

   if 2.6000000000000001e-8 < y

   1. Initial program 99.5%

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

      1. associate-*r/ 99.7%

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

      2. clear-num 98.4%

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

      3. *-commutative 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in y around 0 7.3%

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

      1. clear-num 7.3%

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

      2. *-inverses 7.3%

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

      3. associate-/l* 7.2%

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

      4. *-commutative 7.2%

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

      5. *-un-lft-identity 7.2%

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

      6. times-frac 25.3%

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

      7. /-rgt-identity 25.3%

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

   7. Applied egg-rr 25.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.1% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.0005) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0005) {
		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 <= 0.0005d0) 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 <= 0.0005) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0005:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0005)
		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 <= 0.0005)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0005], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000001e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 71.5%

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

   if 5.0000000000000001e-4 < y

   1. Initial program 99.5%

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

      1. associate-*r/ 99.7%

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

      2. clear-num 98.3%

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

      3. *-commutative 98.3%

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in y around 0 4.9%

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

      1. clear-num 4.9%

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

      2. *-inverses 4.9%

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

      3. associate-/l* 4.8%

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

      4. *-commutative 4.8%

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

      5. *-un-lft-identity 4.8%

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

      6. times-frac 23.5%

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

      7. /-rgt-identity 23.5%

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

   7. Applied egg-rr 23.5%

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

      1. clear-num 25.3%

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

      2. div-inv 25.3%

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

   9. Applied egg-rr 25.3%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.4% 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/ 89.0%

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

   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. associate-/r/ 85.2%

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

6. Applied egg-rr 85.2%

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

   1. associate-*l/ 89.0%

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

   2. *-commutative 89.0%

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

   3. clear-num 88.3%

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

   4. div-inv 87.0%

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

   5. associate-/r* 87.5%

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

   6. associate-/r* 87.6%

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

8. Applied egg-rr 87.6%

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

9. Taylor expanded in y around 0 56.1%

   \[\leadsto \frac{\frac{1}{y}}{\frac{\color{blue}{0.16666666666666666 \cdot y + \frac{1}{y}}}{x}} \]

10. Taylor expanded in x around 0 68.1%

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

    1. *-commutative 68.1%

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

    2. distribute-lft-in 68.1%

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

    3. *-commutative 68.1%

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

    4. rgt-mult-inverse 68.3%

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

12. Simplified 68.3%

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

13. Final simplification 68.3%

    \[\leadsto \frac{x}{y \cdot \left(y \cdot 0.16666666666666666\right) + 1} \]
14. 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 55.6%

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

4. Final simplification 55.6%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))