Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 8.1s

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: 56.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 270000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 270000.0) x (* y (/ -1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 270000.0) {
		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 <= 270000.0d0) 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 <= 270000.0) {
		tmp = x;
	} else {
		tmp = y * (-1.0 / (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 270000.0:
		tmp = x
	else:
		tmp = y * (-1.0 / (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 270000.0)
		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 <= 270000.0)
		tmp = x;
	else
		tmp = y * (-1.0 / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 270000.0], x, N[(y * N[(-1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7e5

   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 2.7e5 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 4.4%

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

      1. *-commutative 4.4%

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

      2. associate-/l* 32.4%

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

   7. Applied egg-rr 32.4%

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

      1. clear-num 34.0%

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

      2. associate-/r/ 32.4%

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

   9. Applied egg-rr 32.4%

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

       1. associate-/r/ 34.0%

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

       2. clear-num 32.4%

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

       3. frac-2neg 32.4%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 32.6%

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

       6. sqr-neg 32.6%

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

       7. sqrt-unprod 32.2%

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

       8. add-sqr-sqrt 32.2%

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

       9. clear-num 33.8%

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

       10. frac-2neg 33.8%

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

       11. metadata-eval 33.8%

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

       12. distribute-frac-neg2 33.8%

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

       13. remove-double-neg 33.8%

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

   11. Applied egg-rr 33.8%

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

Alternative 3: 56.9% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.15e-29) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.15e-29) {
		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 <= 2.15d-29) 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 <= 2.15e-29) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.15e-29:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.15e-29)
		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 <= 2.15e-29)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.15e-29], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1499999999999999e-29

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.3%

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

   if 2.1499999999999999e-29 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 10.0%

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

      1. *-commutative 10.0%

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

      2. associate-/l* 36.0%

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

   7. Applied egg-rr 36.0%

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

      1. clear-num 37.5%

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

      2. div-inv 37.5%

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

   9. Applied egg-rr 37.5%

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

Alternative 4: 56.5% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1e+30) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+30) {
		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 <= 1d+30) 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 <= 1e+30) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+30:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+30)
		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 <= 1e+30)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+30], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e30

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.1%

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

   if 1e30 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 4.4%

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

      1. *-commutative 4.4%

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

      2. associate-/l* 33.3%

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

   7. Applied egg-rr 33.3%

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

Alternative 5: 51.4% 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 50.4%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))