Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.3s

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

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

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: 56.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 920:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 920.0)
   (* x (+ 1.0 (* (* y y) -0.16666666666666666)))
   (* 6.0 (/ x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 920.0) {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 * (x / (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 <= 920.0d0) then
        tmp = x * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = 6.0d0 * (x / (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 920.0) {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 * (x / (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 920.0:
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = 6.0 * (x / (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 920.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(6.0 * Float64(x / Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 920.0)
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = 6.0 * (x / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 920.0], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 920

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 65.4%

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

      1. unpow2 65.4%

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

   4. Simplified 65.4%

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

   if 920 < y

   1. Initial program 99.7%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around 0 23.7%

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

      1. unpow2 23.7%

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

   6. Simplified 23.7%

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

   7. Taylor expanded in y around inf 23.7%

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

      1. unpow2 23.7%

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

   9. Simplified 23.7%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 920:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \end{array} \]

Alternative 3: 56.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.45) x (* 6.0 (/ x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.45) {
		tmp = x;
	} else {
		tmp = 6.0 * (x / (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 <= 2.45d0) then
        tmp = x
    else
        tmp = 6.0d0 * (x / (y * y))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if y <= 2.45:
		tmp = x
	else:
		tmp = 6.0 * (x / (y * y))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, 2.45], x, N[(6.0 * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4500000000000002

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 66.1%

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

   if 2.4500000000000002 < y

   1. Initial program 99.7%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around 0 23.4%

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

      1. unpow2 23.4%

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

   6. Simplified 23.4%

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

   7. Taylor expanded in y around inf 23.4%

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

      1. unpow2 23.4%

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

   9. Simplified 23.4%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \end{array} \]

Alternative 4: 63.2% 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. Step-by-step derivation

   1. clear-num 99.7%

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

   2. un-div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around 0 59.8%

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

   1. unpow2 59.8%

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

6. Simplified 59.8%

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

7. Final simplification 59.8%

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

Alternative 5: 56.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1e+20) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+20) {
		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+20) 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+20) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+20:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+20)
		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+20)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+20], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e20

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 64.9%

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

   if 1e20 < 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. Taylor expanded in y around 0 4.0%

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

      1. associate-/l* 23.0%

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

      2. div-inv 23.0%

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

   5. Applied egg-rr 23.0%

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

   6. Taylor expanded in y around 0 21.9%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 6: 56.5% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0004:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.0004) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0004) {
		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 <= 0.0004d0) 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 <= 0.0004) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0004:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0004)
		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 <= 0.0004)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0004], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.00000000000000019e-4

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 66.1%

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

   if 4.00000000000000019e-4 < 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 4.3%

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

      1. associate-/l* 22.2%

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

      2. div-inv 22.2%

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

   5. Applied egg-rr 22.2%

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

      1. un-div-inv 22.2%

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

   7. Applied egg-rr 22.2%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0004:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]

Alternative 7: 51.0% 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 49.7%

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

3. Final simplification 49.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))