Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 4.4s

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. Final simplification 99.8%

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

Alternative 2: 63.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \lor \neg \left(y \leq 1.25 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{1}{\left(\frac{1}{y} + y \cdot 0.16666666666666666\right) \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.95) (not (<= y 1.25e+19)))
   (/ 1.0 (* (+ (/ 1.0 y) (* y 0.16666666666666666)) (/ y x)))
   (* x (+ 1.0 (* -0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.95) || !(y <= 1.25e+19)) {
		tmp = 1.0 / (((1.0 / y) + (y * 0.16666666666666666)) * (y / x));
	} else {
		tmp = x * (1.0 + (-0.16666666666666666 * (y * 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 <= (-1.95d0)) .or. (.not. (y <= 1.25d+19))) then
        tmp = 1.0d0 / (((1.0d0 / y) + (y * 0.16666666666666666d0)) * (y / x))
    else
        tmp = x * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.95) || !(y <= 1.25e+19)) {
		tmp = 1.0 / (((1.0 / y) + (y * 0.16666666666666666)) * (y / x));
	} else {
		tmp = x * (1.0 + (-0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.95) or not (y <= 1.25e+19):
		tmp = 1.0 / (((1.0 / y) + (y * 0.16666666666666666)) * (y / x))
	else:
		tmp = x * (1.0 + (-0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.95) || !(y <= 1.25e+19))
		tmp = Float64(1.0 / Float64(Float64(Float64(1.0 / y) + Float64(y * 0.16666666666666666)) * Float64(y / x)));
	else
		tmp = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.95) || ~((y <= 1.25e+19)))
		tmp = 1.0 / (((1.0 / y) + (y * 0.16666666666666666)) * (y / x));
	else
		tmp = x * (1.0 + (-0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.95], N[Not[LessEqual[y, 1.25e+19]], $MachinePrecision]], N[(1.0 / N[(N[(N[(1.0 / y), $MachinePrecision] + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.94999999999999996 or 1.25e19 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 98.5%

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

      3. *-commutative 98.5%

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

   3. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y \cdot x}}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 98.5%

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

      2. times-frac 98.4%

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

   5. Applied egg-rr 98.4%

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

   6. Taylor expanded in y around 0 21.0%

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

   if -1.94999999999999996 < y < 1.25e19

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.5%

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

      1. unpow2 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \lor \neg \left(y \leq 1.25 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{1}{\left(\frac{1}{y} + y \cdot 0.16666666666666666\right) \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 3: 62.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \lor \neg \left(y \leq 2.7 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.3) (not (<= y 2.7e+22)))
   (/ y (/ y x))
   (* x (+ 1.0 (* -0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.3) || !(y <= 2.7e+22)) {
		tmp = y / (y / x);
	} else {
		tmp = x * (1.0 + (-0.16666666666666666 * (y * 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 <= (-2.3d0)) .or. (.not. (y <= 2.7d+22))) then
        tmp = y / (y / x)
    else
        tmp = x * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.3) || !(y <= 2.7e+22)) {
		tmp = y / (y / x);
	} else {
		tmp = x * (1.0 + (-0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.3) or not (y <= 2.7e+22):
		tmp = y / (y / x)
	else:
		tmp = x * (1.0 + (-0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.3) || !(y <= 2.7e+22))
		tmp = Float64(y / Float64(y / x));
	else
		tmp = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.3) || ~((y <= 2.7e+22)))
		tmp = y / (y / x);
	else
		tmp = x * (1.0 + (-0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.3], N[Not[LessEqual[y, 2.7e+22]], $MachinePrecision]], N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2999999999999998 or 2.7000000000000002e22 < 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. Taylor expanded in y around 0 4.2%

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

      1. associate-/l* 20.0%

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

      2. div-inv 20.0%

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

      3. clear-num 19.5%

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

   5. Applied egg-rr 19.5%

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

      1. clear-num 20.0%

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

      2. un-div-inv 20.0%

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

   7. Applied egg-rr 20.0%

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

   if -2.2999999999999998 < y < 2.7000000000000002e22

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.5%

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

      1. unpow2 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \lor \neg \left(y \leq 2.7 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 4: 61.8% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-8} \lor \neg \left(y \leq 5 \cdot 10^{+27}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.8e-8) (not (<= y 5e+27))) (* y (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.8e-8) || !(y <= 5e+27)) {
		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 <= (-9.8d-8)) .or. (.not. (y <= 5d+27))) 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 <= -9.8e-8) || !(y <= 5e+27)) {
		tmp = y * (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.8e-8) or not (y <= 5e+27):
		tmp = y * (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.8e-8) || !(y <= 5e+27))
		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 <= -9.8e-8) || ~((y <= 5e+27)))
		tmp = y * (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.8e-8], N[Not[LessEqual[y, 5e+27]], $MachinePrecision]], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -9.8000000000000004e-8 or 4.99999999999999979e27 < 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. Taylor expanded in y around 0 4.9%

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

      1. associate-/l* 20.8%

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

      2. div-inv 20.8%

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

      3. clear-num 20.3%

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

   5. Applied egg-rr 20.3%

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

   if -9.8000000000000004e-8 < y < 4.99999999999999979e27

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.9%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-8} \lor \neg \left(y \leq 5 \cdot 10^{+27}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 62.4% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-8} \lor \neg \left(y \leq 2 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.6e-8) (not (<= y 2e-50))) (/ y (/ y x)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.6e-8) || !(y <= 2e-50)) {
		tmp = y / (y / x);
	} 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 <= (-3.6d-8)) .or. (.not. (y <= 2d-50))) then
        tmp = y / (y / x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.6e-8) || !(y <= 2e-50)) {
		tmp = y / (y / x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.6e-8) or not (y <= 2e-50):
		tmp = y / (y / x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.6e-8) || !(y <= 2e-50))
		tmp = Float64(y / Float64(y / x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.6e-8) || ~((y <= 2e-50)))
		tmp = y / (y / x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.6e-8], N[Not[LessEqual[y, 2e-50]], $MachinePrecision]], N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.59999999999999981e-8 or 2.00000000000000002e-50 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.6%

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

   3. Taylor expanded in y around 0 12.0%

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

      1. associate-/l* 26.3%

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

      2. div-inv 26.3%

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

      3. clear-num 25.9%

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

   5. Applied egg-rr 25.9%

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

      1. clear-num 26.3%

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

      2. un-div-inv 26.3%

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

   7. Applied egg-rr 26.3%

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

   if -3.59999999999999981e-8 < y < 2.00000000000000002e-50

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-8} \lor \neg \left(y \leq 2 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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. Taylor expanded in y around 0 50.6%

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

3. Final simplification 50.6%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023195 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))