Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.8s

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.6e-6) x (/ y (* y (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-6) {
		tmp = x;
	} else {
		tmp = y / (y * (1.0 / 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 <= 1.6d-6) then
        tmp = x
    else
        tmp = y / (y * (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-6) {
		tmp = x;
	} else {
		tmp = y / (y * (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.6e-6:
		tmp = x
	else:
		tmp = y / (y * (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-6)
		tmp = x;
	else
		tmp = Float64(y / Float64(y * Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.6e-6)
		tmp = x;
	else
		tmp = y / (y * (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.6e-6], x, N[(y / N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5999999999999999e-6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.1%

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

   if 1.5999999999999999e-6 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.3%

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

      2. inv-pow 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. times-frac 99.3%

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

      5. unpow-prod-down 99.4%

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

      6. inv-pow 99.4%

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

      7. clear-num 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. frac-times 99.4%

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

      3. *-un-lft-identity 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in y around 0 18.4%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.3e-12) x (* y (/ 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-12) {
		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 <= 3.3d-12) 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 <= 3.3e-12) {
		tmp = x;
	} else {
		tmp = y * (1.0 / (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3e-12:
		tmp = x
	else:
		tmp = y * (1.0 / (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-12)
		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 <= 3.3e-12)
		tmp = x;
	else
		tmp = y * (1.0 / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3e-12], x, N[(y * N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3000000000000001e-12

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.0%

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

   if 3.3000000000000001e-12 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 18.2%

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

      1. associate-*r/ 6.6%

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

      2. clear-num 6.6%

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

      3. associate-/r* 6.8%

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

      4. *-un-lft-identity 6.8%

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

      5. associate-*l/ 6.8%

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

      6. lft-mult-inverse 6.8%

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

   8. Applied egg-rr 6.8%

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

      1. *-inverses 6.8%

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

      2. associate-/r* 19.6%

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

      3. div-inv 19.6%

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

      4. *-un-lft-identity 19.6%

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

      5. associate-*l/ 19.6%

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

      6. frac-2neg 19.6%

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

      7. div-inv 19.6%

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

      8. associate-*l/ 19.6%

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

      9. *-un-lft-identity 19.6%

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

      10. distribute-neg-frac2 19.6%

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

   10. Applied egg-rr 19.6%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1e-12) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e-12) {
		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-12) 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-12) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e-12:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e-12)
		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-12)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e-12], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999998e-13

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.0%

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

   if 9.9999999999999998e-13 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 18.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.1% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.4e-6) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-6) {
		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 <= 3.4d-6) 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 <= 3.4e-6) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.4e-6:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-6)
		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 <= 3.4e-6)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.4e-6], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.40000000000000006e-6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.1%

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

   if 3.40000000000000006e-6 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 17.0%

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

      1. clear-num 18.4%

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

      2. un-div-inv 18.4%

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

   8. Applied egg-rr 18.4%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]
5. 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 50.7%

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

4. Final simplification 50.7%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))