Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.2s

Alternatives: 7

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

\[\begin{array}{l} \\ \frac{x}{\frac{y}{\sin y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x (/ y (sin y))))

C

double code(double x, double y) {
	return x / (y / sin(y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y / sin(y))
end function

Java

public static double code(double x, double y) {
	return x / (y / Math.sin(y));
}

Python

def code(x, y):
	return x / (y / math.sin(y))

Julia

function code(x, y)
	return Float64(x / Float64(y / sin(y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (y / sin(y));
end

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in x around 0 88.3%

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

   1. *-commutative 88.3%

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

   2. associate-/l* 99.8%

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

4. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 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. Final simplification 99.8%

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

Alternative 3: 63.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -450000 \lor \neg \left(y \leq 3300\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -450000.0) (not (<= y 3300.0)))
   (* 6.0 (/ x (* y y)))
   (* x (+ 1.0 (* (* y y) -0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -450000.0) || !(y <= 3300.0)) {
		tmp = 6.0 * (x / (y * y));
	} else {
		tmp = x * (1.0 + ((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 <= (-450000.0d0)) .or. (.not. (y <= 3300.0d0))) then
        tmp = 6.0d0 * (x / (y * y))
    else
        tmp = x * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -450000.0) || !(y <= 3300.0)) {
		tmp = 6.0 * (x / (y * y));
	} else {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -450000.0) or not (y <= 3300.0):
		tmp = 6.0 * (x / (y * y))
	else:
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -450000.0) || !(y <= 3300.0))
		tmp = Float64(6.0 * Float64(x / Float64(y * y)));
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -450000.0) || ~((y <= 3300.0)))
		tmp = 6.0 * (x / (y * y));
	else
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -450000.0], N[Not[LessEqual[y, 3300.0]], $MachinePrecision]], N[(6.0 * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e5 or 3300 < y

   1. Initial program 99.7%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around 0 24.1%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 24.1%

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

   7. Simplified 24.1%

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

   8. Taylor expanded in y around inf 24.1%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 24.1%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

   10. Simplified 24.1%

       \[\leadsto \color{blue}{6 \cdot \frac{x}{y \cdot y}} \]

   if -4.5e5 < y < 3300

   1. Initial program 100.0%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 98.2%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 98.2%

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

   4. Simplified 98.2%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -450000 \lor \neg \left(y \leq 3300\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 4: 63.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.45\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.5) (not (<= y 2.45))) (* 6.0 (/ x (* y y))) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.45)) {
		tmp = 6.0 * (x / (y * y));
	} else {
		tmp = 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 <= (-2.5d0)) .or. (.not. (y <= 2.45d0))) then
        tmp = 6.0d0 * (x / (y * y))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.45)) {
		tmp = 6.0 * (x / (y * y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.5) or not (y <= 2.45):
		tmp = 6.0 * (x / (y * y))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.5) || !(y <= 2.45))
		tmp = Float64(6.0 * Float64(x / Float64(y * y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.5) || ~((y <= 2.45)))
		tmp = 6.0 * (x / (y * y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.5], N[Not[LessEqual[y, 2.45]], $MachinePrecision]], N[(6.0 * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.5 or 2.4500000000000002 < y

   1. Initial program 99.6%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around 0 23.8%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 23.8%

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

   7. Simplified 23.8%

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

   8. Taylor expanded in y around inf 23.8%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 23.8%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

   10. Simplified 23.8%

       \[\leadsto \color{blue}{6 \cdot \frac{x}{y \cdot y}} \]

   if -2.5 < y < 2.4500000000000002

   1. Initial program 100.0%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 99.1%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.45\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 62.1% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+54} \lor \neg \left(y \leq 5 \cdot 10^{+21}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e+54) (not (<= y 5e+21))) (* y (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5e+54) || !(y <= 5e+21)) {
		tmp = y * (x / y);
	} else {
		tmp = 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 <= (-5d+54)) .or. (.not. (y <= 5d+21))) then
        tmp = y * (x / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5e+54) || !(y <= 5e+21)) {
		tmp = y * (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5e+54) or not (y <= 5e+21):
		tmp = y * (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5e+54) || !(y <= 5e+21))
		tmp = Float64(y * Float64(x / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5e+54) || ~((y <= 5e+21)))
		tmp = y * (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5e+54], N[Not[LessEqual[y, 5e+21]], $MachinePrecision]], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000005e54 or 5e21 < y

   1. Initial program 99.6%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in y around 0 4.1%

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

      1. *-un-lft-identity 4.1%

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

      2. times-frac 25.6%

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

      3. /-rgt-identity 25.6%

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

   5. Applied egg-rr 25.6%

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

   if -5.00000000000000005e54 < y < 5e21

   1. Initial program 100.0%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 89.2%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+54} \lor \neg \left(y \leq 5 \cdot 10^{+21}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 63.7% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in x around 0 88.3%

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

   1. *-commutative 88.3%

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

   2. associate-/l* 99.8%

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

4. Simplified 99.8%

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

5. Taylor expanded in y around 0 64.0%

   \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
6. Step-by-step derivation

   1. unpow2 64.0%

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

7. Simplified 64.0%

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

8. Final simplification 64.0%

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

Alternative 7: 51.8% 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. Taylor expanded in y around 0 54.7%

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

3. Final simplification 54.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023185 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))