Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.0s

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: 57.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{-y}{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.2e+40) x (/ 1.0 (/ (/ (- y) x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.2e+40) {
		tmp = x;
	} else {
		tmp = 1.0 / ((-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 <= 4.2d+40) then
        tmp = x
    else
        tmp = 1.0d0 / ((-y / x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.2e+40) {
		tmp = x;
	} else {
		tmp = 1.0 / ((-y / x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.2e+40:
		tmp = x
	else:
		tmp = 1.0 / ((-y / x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.2e+40)
		tmp = x;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(-y) / x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.2e+40)
		tmp = x;
	else
		tmp = 1.0 / ((-y / x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.2e+40], x, N[(1.0 / N[(N[((-y) / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2000000000000002e40

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 63.7%

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

   if 4.2000000000000002e40 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in y around 0 4.1%

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

      1. associate-/r* 29.6%

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

      2. div-inv 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. frac-2neg 29.6%

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

      2. metadata-eval 29.6%

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

      3. associate-*r/ 29.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 11.7%

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

      6. sqr-neg 11.7%

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

      7. sqrt-unprod 29.8%

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

      8. add-sqr-sqrt 29.8%

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

   9. Applied egg-rr 29.8%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{-y}{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.9e-17) x (* y (/ 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.9e-17) {
		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 <= 2.9d-17) 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 <= 2.9e-17) {
		tmp = x;
	} else {
		tmp = y * (1.0 / (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.9e-17:
		tmp = x
	else:
		tmp = y * (1.0 / (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.9e-17)
		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 <= 2.9e-17)
		tmp = x;
	else
		tmp = y * (1.0 / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.9e-17], x, N[(y * N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9000000000000003e-17

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.3%

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

   if 2.9000000000000003e-17 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in y around 0 8.2%

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

      1. clear-num 8.2%

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

      2. *-inverses 8.2%

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

      3. associate-/l* 8.1%

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

      4. clear-num 8.1%

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

      5. associate-/r* 29.3%

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

      6. associate-/r/ 29.3%

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

   7. Applied egg-rr 29.3%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5e+26) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+26) {
		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 <= 5d+26) 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 <= 5e+26) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+26:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+26)
		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 <= 5e+26)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+26], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000001e26

   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 5.0000000000000001e26 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in y around 0 4.3%

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

      1. clear-num 4.3%

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

      2. *-inverses 4.3%

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

      3. associate-/r/ 27.4%

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

   7. Applied egg-rr 27.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.3% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2e-8) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-8) {
		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 <= 2d-8) 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 <= 2e-8) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-8:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-8], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.7%

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

   if 2e-8 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in y around 0 5.7%

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

      1. clear-num 5.7%

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

      2. *-inverses 5.7%

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

      3. associate-/r/ 27.0%

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

   7. Applied egg-rr 27.0%

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

      1. clear-num 27.4%

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

      2. associate-*l/ 27.4%

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

      3. *-un-lft-identity 27.4%

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

   9. Applied egg-rr 27.4%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 48.8%

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

4. Final simplification 48.8%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))