Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.9s

Alternatives: 8

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. Step-by-step derivation

   1. associate-*r/ 91.4%

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

   2. associate-/l* 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]
4. Add Preprocessing

5. Final simplification 99.8%

   \[\leadsto \frac{x}{\frac{y}{\sin y}} \]
6. Add Preprocessing

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. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 3: 58.7% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(6 \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4) x (* (/ 1.0 y) (* 6.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (6.0 * (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 <= 2.4d0) then
        tmp = x
    else
        tmp = (1.0d0 / y) * (6.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (6.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4:
		tmp = x
	else:
		tmp = (1.0 / y) * (6.0 * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4)
		tmp = x;
	else
		tmp = Float64(Float64(1.0 / y) * Float64(6.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x;
	else
		tmp = (1.0 / y) * (6.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(N[(1.0 / y), $MachinePrecision] * N[(6.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 67.7%

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

   if 2.39999999999999991 < 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. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around 0 29.4%

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

   8. Taylor expanded in y around inf 29.4%

      \[\leadsto \frac{x}{\color{blue}{\left(0.16666666666666666 \cdot y\right)} \cdot y} \]
   9. Step-by-step derivation

      1. *-commutative 29.4%

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

   10. Simplified 29.4%

       \[\leadsto \frac{x}{\color{blue}{\left(y \cdot 0.16666666666666666\right)} \cdot y} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 29.4%

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

       2. lft-mult-inverse 29.4%

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

       3. associate-*l/ 29.4%

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

       4. *-un-lft-identity 29.4%

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

       5. associate-/r/ 29.4%

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

       6. *-un-lft-identity 29.4%

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

       7. *-commutative 29.4%

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

       8. times-frac 29.5%

          \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{\frac{y}{\frac{y}{x}}}{y \cdot 0.16666666666666666}} \]

       9. *-un-lft-identity 29.5%

          \[\leadsto \frac{1}{y} \cdot \frac{\color{blue}{1 \cdot \frac{y}{\frac{y}{x}}}}{y \cdot 0.16666666666666666} \]

       10. *-commutative 29.5%

           \[\leadsto \frac{1}{y} \cdot \frac{1 \cdot \frac{y}{\frac{y}{x}}}{\color{blue}{0.16666666666666666 \cdot y}} \]

       11. times-frac 29.5%

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

       12. associate-/r/ 29.5%

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

       13. *-un-lft-identity 29.5%

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

       14. associate-*l/ 29.5%

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

       15. lft-mult-inverse 29.5%

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

       16. *-un-lft-identity 29.5%

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

       17. metadata-eval 29.5%

           \[\leadsto \frac{1}{y} \cdot \left(\color{blue}{6} \cdot \frac{x}{y}\right) \]

   12. Applied egg-rr 29.5%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(6 \cdot \frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(6 \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.4% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*r/ 91.4%

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

   2. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.8%

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

   2. associate-/r/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in y around 0 63.6%

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

8. Final simplification 63.6%

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

Alternative 5: 58.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4) x (/ x (* y (* y 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = x / (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 <= 2.4d0) then
        tmp = x
    else
        tmp = x / (y * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = x / (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4:
		tmp = x
	else:
		tmp = x / (y * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4)
		tmp = x;
	else
		tmp = Float64(x / Float64(y * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x;
	else
		tmp = x / (y * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(x / N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 67.7%

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

   if 2.39999999999999991 < 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. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around 0 29.4%

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

   8. Taylor expanded in y around inf 29.4%

      \[\leadsto \frac{x}{\color{blue}{\left(0.16666666666666666 \cdot y\right)} \cdot y} \]
   9. Step-by-step derivation

      1. *-commutative 29.4%

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

   10. Simplified 29.4%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.0% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+61}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1e+61) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+61) {
		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+61) 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+61) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+61:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+61)
		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+61)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+61], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999949e60

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 63.3%

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

   if 9.99999999999999949e60 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l/ 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0 4.0%

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

      1. *-commutative 4.0%

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

   7. Simplified 4.0%

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

      1. *-un-lft-identity 4.0%

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

      2. times-frac 28.6%

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

      3. /-rgt-identity 28.6%

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

      4. *-commutative 28.6%

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

   9. Applied egg-rr 28.6%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+61}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.4% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.002) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.002) {
		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 <= 0.002d0) 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 <= 0.002) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.002:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.002)
		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 <= 0.002)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.002], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 67.9%

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

   if 2e-3 < 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. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

      1. associate-/l* 99.6%

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

      2. div-inv 99.5%

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

      3. *-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around 0 4.8%

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

      1. *-commutative 4.8%

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

   9. Simplified 4.8%

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

       1. div-inv 4.8%

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

       2. associate-/l* 28.5%

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

   11. Applied egg-rr 28.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.6% 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 49.2%

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

4. Final simplification 49.2%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))