Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.8s

Alternatives: 4

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 99.7%

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

Alternative 2: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 4 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (/ (sin y) y) 4e-84) 0.0 x))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 4e-84) {
		tmp = 0.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((sin(y) / y) <= 4d-84) then
        tmp = 0.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sin(y) / y) <= 4e-84) {
		tmp = 0.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(y) / y) <= 4e-84:
		tmp = 0.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 4e-84)
		tmp = 0.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(y) / y) <= 4e-84)
		tmp = 0.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 4e-84], 0.0, x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 4.0000000000000001e-84

   1. Initial program 99.5%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-+PI-rev N/A

         \[\leadsto x \cdot \frac{\mathsf{neg}\left(\color{blue}{\sin \left(y + \mathsf{PI}\left(\right)\right)}\right)}{y} \]

      4. sin-neg-rev N/A

         \[\leadsto x \cdot \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\left(y + \mathsf{PI}\left(\right)\right)\right)\right)}}{y} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{PI}\left(\right) + y\right)}\right)\right)}{y} \]

      6. distribute-neg-in N/A

         \[\leadsto x \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)}}{y} \]

      7. sin-sum N/A

         \[\leadsto x \cdot \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \cos \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}{y} \]

      8. cos-neg-rev N/A

         \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\cos y} + \cos \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}{y} \]

      9. cos-neg-rev N/A

         \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\cos \mathsf{PI}\left(\right)} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}{y} \]

      10. sin-neg-rev N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \cos \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right)}}{y} \]

      11. lift-sin.f64 N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \cos \mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin y}\right)\right)}{y} \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\left(\mathsf{neg}\left(\cos \mathsf{PI}\left(\right) \cdot \sin y\right)\right)}}{y} \]

      13. *-commutative N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \left(\mathsf{neg}\left(\color{blue}{\sin y \cdot \cos \mathsf{PI}\left(\right)}\right)\right)}{y} \]

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

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \cos \mathsf{PI}\left(\right)}}{y} \]

      15. lift-sin.f64 N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \left(\mathsf{neg}\left(\color{blue}{\sin y}\right)\right) \cdot \cos \mathsf{PI}\left(\right)}{y} \]

      16. sin-neg-rev N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)} \cdot \cos \mathsf{PI}\left(\right)}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right), \cos y, \sin \left(\mathsf{neg}\left(y\right)\right) \cdot \cos \mathsf{PI}\left(\right)\right)}}{y} \]

   4. Applied rewrites 98.5%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\sin \left(-\mathsf{PI}\left(\right)\right), \cos y, 1 \cdot \sin y\right)}}{y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot \sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sin-neg N/A

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

      4. sin-PI N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{0}\right)\right) \cdot \frac{x}{y} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{0} \cdot \frac{x}{y} \]

      6. mul0-lft 26.0

         \[\leadsto \color{blue}{0} \]

   7. Applied rewrites 26.0%

      \[\leadsto \color{blue}{0} \]

   if 4.0000000000000001e-84 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-+PI-rev N/A

         \[\leadsto x \cdot \frac{\mathsf{neg}\left(\color{blue}{\sin \left(y + \mathsf{PI}\left(\right)\right)}\right)}{y} \]

      4. sin-neg-rev N/A

         \[\leadsto x \cdot \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\left(y + \mathsf{PI}\left(\right)\right)\right)\right)}}{y} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{PI}\left(\right) + y\right)}\right)\right)}{y} \]

      6. distribute-neg-in N/A

         \[\leadsto x \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)}}{y} \]

      7. sin-sum N/A

         \[\leadsto x \cdot \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \cos \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}{y} \]

      8. cos-neg-rev N/A

         \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\cos y} + \cos \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}{y} \]

      9. cos-neg-rev N/A

         \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\cos \mathsf{PI}\left(\right)} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}{y} \]

      10. sin-neg-rev N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \cos \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right)}}{y} \]

      11. lift-sin.f64 N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \cos \mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin y}\right)\right)}{y} \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\left(\mathsf{neg}\left(\cos \mathsf{PI}\left(\right) \cdot \sin y\right)\right)}}{y} \]

      13. *-commutative N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \left(\mathsf{neg}\left(\color{blue}{\sin y \cdot \cos \mathsf{PI}\left(\right)}\right)\right)}{y} \]

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

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \cos \mathsf{PI}\left(\right)}}{y} \]

      15. lift-sin.f64 N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \left(\mathsf{neg}\left(\color{blue}{\sin y}\right)\right) \cdot \cos \mathsf{PI}\left(\right)}{y} \]

      16. sin-neg-rev N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)} \cdot \cos \mathsf{PI}\left(\right)}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right), \cos y, \sin \left(\mathsf{neg}\left(y\right)\right) \cdot \cos \mathsf{PI}\left(\right)\right)}}{y} \]

   4. Applied rewrites 16.4%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\sin \left(-\mathsf{PI}\left(\right)\right), \cos y, 1 \cdot \sin y\right)}}{y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot y + x \cdot \sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{y}} \]
   6. Step-by-step derivation

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y} + \frac{x \cdot \sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot y}{y} + \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot x}}{y} \]

      3. associate-/l* N/A

         \[\leadsto \frac{x \cdot y}{y} + \color{blue}{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \frac{x}{y}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{y} + \sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \frac{x}{y} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{y}} + \sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \frac{x}{y} \]

      6. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \left(y + \sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)\right)} \]

      7. sin-neg N/A

         \[\leadsto \frac{x}{y} \cdot \left(y + \color{blue}{\left(\mathsf{neg}\left(\sin \mathsf{PI}\left(\right)\right)\right)}\right) \]

      8. sin-PI N/A

         \[\leadsto \frac{x}{y} \cdot \left(y + \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{x}{y} \cdot \left(y + \color{blue}{0}\right) \]

      10. +-rgt-identity N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. *-inverses N/A

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

      16. *-rgt-identity 90.2

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

   7. Applied rewrites 90.2%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 4 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \left(-0.16666666666666666 \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.05e+29) (fma y (* y (* -0.16666666666666666 x)) x) 0.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.05e+29) {
		tmp = fma(y, (y * (-0.16666666666666666 * x)), x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.05e+29)
		tmp = fma(y, Float64(y * Float64(-0.16666666666666666 * x)), x);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.05e+29], N[(y * N[(y * N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if y < 2.0500000000000002e29

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. remove-double-neg N/A

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

      12. lower-*.f64 79.8

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 64.1

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

   7. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, -0.16666666666666666, x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.1%

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

   if 2.0500000000000002e29 < y

   1. Initial program 99.4%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-+PI-rev N/A

         \[\leadsto x \cdot \frac{\mathsf{neg}\left(\color{blue}{\sin \left(y + \mathsf{PI}\left(\right)\right)}\right)}{y} \]

      4. sin-neg-rev N/A

         \[\leadsto x \cdot \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\left(y + \mathsf{PI}\left(\right)\right)\right)\right)}}{y} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{PI}\left(\right) + y\right)}\right)\right)}{y} \]

      6. distribute-neg-in N/A

         \[\leadsto x \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)}}{y} \]

      7. sin-sum N/A

         \[\leadsto x \cdot \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \cos \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}{y} \]

      8. cos-neg-rev N/A

         \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\cos y} + \cos \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}{y} \]

      9. cos-neg-rev N/A

         \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\cos \mathsf{PI}\left(\right)} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}{y} \]

      10. sin-neg-rev N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \cos \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right)}}{y} \]

      11. lift-sin.f64 N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \cos \mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin y}\right)\right)}{y} \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\left(\mathsf{neg}\left(\cos \mathsf{PI}\left(\right) \cdot \sin y\right)\right)}}{y} \]

      13. *-commutative N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \left(\mathsf{neg}\left(\color{blue}{\sin y \cdot \cos \mathsf{PI}\left(\right)}\right)\right)}{y} \]

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

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \cos \mathsf{PI}\left(\right)}}{y} \]

      15. lift-sin.f64 N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \left(\mathsf{neg}\left(\color{blue}{\sin y}\right)\right) \cdot \cos \mathsf{PI}\left(\right)}{y} \]

      16. sin-neg-rev N/A

          \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)} \cdot \cos \mathsf{PI}\left(\right)}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right), \cos y, \sin \left(\mathsf{neg}\left(y\right)\right) \cdot \cos \mathsf{PI}\left(\right)\right)}}{y} \]

   4. Applied rewrites 98.6%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\sin \left(-\mathsf{PI}\left(\right)\right), \cos y, 1 \cdot \sin y\right)}}{y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot \sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sin-neg N/A

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

      4. sin-PI N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{0}\right)\right) \cdot \frac{x}{y} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{0} \cdot \frac{x}{y} \]

      6. mul0-lft 25.3

         \[\leadsto \color{blue}{0} \]

   7. Applied rewrites 25.3%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \left(-0.16666666666666666 \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 16.4% accurate, 117.0× 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 99.7%

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

   1. remove-double-neg N/A

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

   2. lift-sin.f64 N/A

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

   3. sin-+PI-rev N/A

      \[\leadsto x \cdot \frac{\mathsf{neg}\left(\color{blue}{\sin \left(y + \mathsf{PI}\left(\right)\right)}\right)}{y} \]

   4. sin-neg-rev N/A

      \[\leadsto x \cdot \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\left(y + \mathsf{PI}\left(\right)\right)\right)\right)}}{y} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{PI}\left(\right) + y\right)}\right)\right)}{y} \]

   6. distribute-neg-in N/A

      \[\leadsto x \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)}}{y} \]

   7. sin-sum N/A

      \[\leadsto x \cdot \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \cos \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}{y} \]

   8. cos-neg-rev N/A

      \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\cos y} + \cos \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}{y} \]

   9. cos-neg-rev N/A

      \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\cos \mathsf{PI}\left(\right)} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}{y} \]

   10. sin-neg-rev N/A

       \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \cos \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right)}}{y} \]

   11. lift-sin.f64 N/A

       \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \cos \mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin y}\right)\right)}{y} \]

   12. distribute-rgt-neg-in N/A

       \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\left(\mathsf{neg}\left(\cos \mathsf{PI}\left(\right) \cdot \sin y\right)\right)}}{y} \]

   13. *-commutative N/A

       \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \left(\mathsf{neg}\left(\color{blue}{\sin y \cdot \cos \mathsf{PI}\left(\right)}\right)\right)}{y} \]

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

       \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \cos \mathsf{PI}\left(\right)}}{y} \]

   15. lift-sin.f64 N/A

       \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \left(\mathsf{neg}\left(\color{blue}{\sin y}\right)\right) \cdot \cos \mathsf{PI}\left(\right)}{y} \]

   16. sin-neg-rev N/A

       \[\leadsto x \cdot \frac{\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \cos y + \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)} \cdot \cos \mathsf{PI}\left(\right)}{y} \]

   17. lower-fma.f64 N/A

       \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right), \cos y, \sin \left(\mathsf{neg}\left(y\right)\right) \cdot \cos \mathsf{PI}\left(\right)\right)}}{y} \]

4. Applied rewrites 55.2%

   \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\sin \left(-\mathsf{PI}\left(\right)\right), \cos y, 1 \cdot \sin y\right)}}{y} \]

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x \cdot \sin \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{y}} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. sin-neg N/A

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

   4. sin-PI N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{0}\right)\right) \cdot \frac{x}{y} \]

   5. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot \frac{x}{y} \]

   6. mul0-lft 14.6

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 14.6%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 14.6%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))