Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 56.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 260:\\ \;\;\;\;x + x \cdot \left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 6}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 260.0)
   (+ x (* x (* -0.16666666666666666 (* y y))))
   (/ (/ (* x 6.0) y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 260.0) {
		tmp = x + (x * (-0.16666666666666666 * (y * y)));
	} else {
		tmp = ((x * 6.0) / 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 <= 260.0d0) then
        tmp = x + (x * ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = ((x * 6.0d0) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 260.0) {
		tmp = x + (x * (-0.16666666666666666 * (y * y)));
	} else {
		tmp = ((x * 6.0) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 260.0:
		tmp = x + (x * (-0.16666666666666666 * (y * y)))
	else:
		tmp = ((x * 6.0) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 260.0)
		tmp = Float64(x + Float64(x * Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(Float64(Float64(x * 6.0) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 260.0)
		tmp = x + (x * (-0.16666666666666666 * (y * y)));
	else
		tmp = ((x * 6.0) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 260.0], N[(x + N[(x * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 6.0), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 260

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

      3. associate-*l* N/A

         \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 68.6%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot x\right), \color{blue}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right), x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{-1}{6} \cdot \left(y \cdot y\right)\right)\right), x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \left(y \cdot y\right)\right)\right), x\right) \]

      10. *-lowering-*.f64 68.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), x\right) \]

   7. Applied egg-rr 68.6%

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

   if 260 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

      3. associate-*l* N/A

         \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 2.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 2.2%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-neg-in N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)} \cdot y\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}} \cdot y\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right), \color{blue}{y}\right)\right) \]

   7. Applied egg-rr 2.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}, y\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right), y\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right), y\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot y\right) \cdot y\right)\right), y\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)\right), y\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot y\right)\right)\right), y\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right), y\right)\right) \]

      7. *-lowering-*.f64 32.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right), y\right)\right) \]

   10. Simplified 32.1%

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

   11. Taylor expanded in y around inf

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

       1. associate-*r/ N/A

          \[\leadsto \frac{6 \cdot x}{\color{blue}{{y}^{2}}} \]

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{6 \cdot x}{y}\right), \color{blue}{y}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(6 \cdot x\right), y\right), y\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot 6\right), y\right), y\right) \]

       7. *-lowering-*.f64 32.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), y\right), y\right) \]

   13. Simplified 32.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 260:\\ \;\;\;\;x + x \cdot \left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 6}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 260:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 6}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 260.0)
   (* x (+ 1.0 (* y (* y -0.16666666666666666))))
   (/ (/ (* x 6.0) y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 260.0) {
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = ((x * 6.0) / 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 <= 260.0d0) then
        tmp = x * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
    else
        tmp = ((x * 6.0d0) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 260.0) {
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = ((x * 6.0) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 260.0:
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)))
	else:
		tmp = ((x * 6.0) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 260.0)
		tmp = Float64(x * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))));
	else
		tmp = Float64(Float64(Float64(x * 6.0) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 260.0)
		tmp = x * (1.0 + (y * (y * -0.16666666666666666)));
	else
		tmp = ((x * 6.0) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 260.0], N[(x * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 6.0), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 260

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

      3. associate-*l* N/A

         \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 68.6%

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

   if 260 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

      3. associate-*l* N/A

         \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 2.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 2.2%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-neg-in N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)} \cdot y\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}} \cdot y\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right), \color{blue}{y}\right)\right) \]

   7. Applied egg-rr 2.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}, y\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right), y\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right), y\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot y\right) \cdot y\right)\right), y\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)\right), y\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot y\right)\right)\right), y\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right), y\right)\right) \]

      7. *-lowering-*.f64 32.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right), y\right)\right) \]

   10. Simplified 32.1%

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

   11. Taylor expanded in y around inf

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

       1. associate-*r/ N/A

          \[\leadsto \frac{6 \cdot x}{\color{blue}{{y}^{2}}} \]

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{6 \cdot x}{y}\right), \color{blue}{y}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(6 \cdot x\right), y\right), y\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot 6\right), y\right), y\right) \]

       7. *-lowering-*.f64 32.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), y\right), y\right) \]

   13. Simplified 32.9%

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

Alternative 4: 56.4% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   5. Simplified 68.6%

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

   if 2.39999999999999991 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

      3. associate-*l* N/A

         \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 2.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 2.2%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-neg-in N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)} \cdot y\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}} \cdot y\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right), \color{blue}{y}\right)\right) \]

   7. Applied egg-rr 2.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}, y\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right), y\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right), y\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot y\right) \cdot y\right)\right), y\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)\right), y\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot y\right)\right)\right), y\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right), y\right)\right) \]

      7. *-lowering-*.f64 32.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right), y\right)\right) \]

   10. Simplified 32.1%

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

   11. Taylor expanded in y around inf

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

       1. associate-*r/ N/A

          \[\leadsto \frac{6 \cdot x}{\color{blue}{{y}^{2}}} \]

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{6 \cdot x}{y}\right), \color{blue}{y}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(6 \cdot x\right), y\right), y\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot 6\right), y\right), y\right) \]

       7. *-lowering-*.f64 32.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), y\right), y\right) \]

   13. Simplified 32.9%

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

Alternative 5: 62.6% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto x \cdot 1 + \color{blue}{\frac{-1}{6}} \cdot \left(x \cdot {y}^{2}\right) \]

   2. *-commutative N/A

      \[\leadsto x \cdot 1 + \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{6}} \]

   3. associate-*l* N/A

      \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} \]

   4. *-commutative N/A

      \[\leadsto x \cdot 1 + x \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

   5. distribute-lft-in N/A

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 54.3%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

5. Simplified 54.3%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. *-rgt-identity N/A

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

   4. *-inverses N/A

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft-neg-in N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)} \cdot y\right)}\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}} \cdot y\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(\frac{1 - y \cdot \left(y \cdot \frac{-1}{6}\right)}{1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}\right), \color{blue}{y}\right)\right) \]

7. Applied egg-rr 40.6%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}, y\right)\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right), y\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right), y\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot y\right) \cdot y\right)\right), y\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)\right), y\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot y\right)\right)\right), y\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right), y\right)\right) \]

   7. *-lowering-*.f64 51.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right), y\right)\right) \]

10. Simplified 51.9%

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

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{x}{1 + \frac{1}{6} \cdot {y}^{2}}} \]
12. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {\color{blue}{y}}^{2}\right)\right) \]

    3. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

    4. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(1 - \color{blue}{\frac{-1}{6} \cdot {y}^{2}}\right)\right) \]

    5. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \left(y \cdot \left(\frac{-1}{6} \cdot \color{blue}{y}\right)\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{6} \cdot y\right)}\right)\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

    12. *-lowering-*.f64 66.0%

        \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

13. Simplified 66.0%

    \[\leadsto \color{blue}{\frac{x}{1 - y \cdot \left(y \cdot -0.16666666666666666\right)}} \]
14. Add Preprocessing

Alternative 6: 56.3% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.8e+18) x 0.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.8e+18) {
		tmp = x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.8d+18) then
        tmp = x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.8e+18) {
		tmp = x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.8e+18:
		tmp = x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.8e+18)
		tmp = x;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.8e+18)
		tmp = x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.8e+18], x, 0.0]

Derivation

1. Split input into 2 regimes

2. if y < 5.8e18

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 67.8%

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

   if 5.8e18 < y

   1. Initial program 99.6%

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

   4. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 15.7% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 14.6%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))