Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.1s

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

\[\begin{array}{l} \\ x \cdot \frac{\sin y}{y} \end{array} \]

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 99.9%

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

Alternative 2: 63.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-21}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(\left(0.008333333333333333 \cdot y\right) \cdot y\right) \cdot y, y, \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sin y) y) 1e-21)
   0.0
   (*
    x
    (fma
     (* (* (* 0.008333333333333333 y) y) y)
     y
     (fma (* -0.16666666666666666 y) y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 1e-21) {
		tmp = 0.0;
	} else {
		tmp = x * fma((((0.008333333333333333 * y) * y) * y), y, fma((-0.16666666666666666 * y), y, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 1e-21)
		tmp = 0.0;
	else
		tmp = Float64(x * fma(Float64(Float64(Float64(0.008333333333333333 * y) * y) * y), y, fma(Float64(-0.16666666666666666 * y), y, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-21], 0.0, N[(x * N[(N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y + N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 9.99999999999999908e-22

   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 98.7%

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

   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 28.0

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

   7. Applied rewrites 28.0%

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

   if 9.99999999999999908e-22 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right) + \frac{-1}{6} \cdot x\right)} + x \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) \cdot {y}^{2} + \left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2}\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)} + \left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2}\right) + x \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right)} \]

      6. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot x\right)}\right) + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      7. associate-*r* N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot x\right)} + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x} + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \left(\color{blue}{\frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} + x\right) \]

      10. *-commutative N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \left(\frac{-1}{6} \cdot \color{blue}{\left({y}^{2} \cdot x\right)} + x\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \left(\color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right) \cdot x} + x\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right) \cdot x} \]

      13. +-commutative N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \cdot x \]

      14. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

      \[\leadsto x \cdot \mathsf{fma}\left(\left(\left(0.008333333333333333 \cdot y\right) \cdot y\right) \cdot y, \color{blue}{y}, \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-21}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(\left(0.008333333333333333 \cdot y\right) \cdot y\right) \cdot y, y, \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-21}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sin y) y) 1e-21)
   0.0
   (*
    x
    (fma
     (* y y)
     (fma (* 0.008333333333333333 y) y -0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 1e-21) {
		tmp = 0.0;
	} else {
		tmp = x * fma((y * y), fma((0.008333333333333333 * y), y, -0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 1e-21)
		tmp = 0.0;
	else
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(0.008333333333333333 * y), y, -0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-21], 0.0, N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 9.99999999999999908e-22

   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 98.7%

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

   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 28.0

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

   7. Applied rewrites 28.0%

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

   if 9.99999999999999908e-22 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right) + \frac{-1}{6} \cdot x\right)} + x \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) \cdot {y}^{2} + \left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2}\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)} + \left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2}\right) + x \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right)} \]

      6. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot x\right)}\right) + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      7. associate-*r* N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot x\right)} + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x} + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \left(\color{blue}{\frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} + x\right) \]

      10. *-commutative N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \left(\frac{-1}{6} \cdot \color{blue}{\left({y}^{2} \cdot x\right)} + x\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \left(\color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right) \cdot x} + x\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right) \cdot x} \]

      13. +-commutative N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \cdot x \]

      14. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-21}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq 5 \cdot 10^{-307}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x (/ (sin y) y)) 5e-307) 0.0 x))

C

double code(double x, double y) {
	double tmp;
	if ((x * (sin(y) / y)) <= 5e-307) {
		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 ((x * (sin(y) / y)) <= 5d-307) 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 ((x * (Math.sin(y) / y)) <= 5e-307) {
		tmp = 0.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * (math.sin(y) / y)) <= 5e-307:
		tmp = 0.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * Float64(sin(y) / y)) <= 5e-307)
		tmp = 0.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * (sin(y) / y)) <= 5e-307)
		tmp = 0.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 5e-307], 0.0, x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < 5.00000000000000014e-307

   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 53.7%

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

   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 20.5

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

   7. Applied rewrites 20.5%

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

   if 5.00000000000000014e-307 < (*.f64 x (/.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 38.4%

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

   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

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      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. +-commutative N/A

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

      8. sin-neg N/A

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

      9. sin-PI N/A

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

      10. metadata-eval N/A

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

      11. +-lft-identity N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. *-inverses N/A

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

      17. *-rgt-identity 66.2

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

   7. Applied rewrites 66.2%

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

Alternative 5: 57.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot x\\ \mathbf{if}\;y \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \left(y \cdot y\right) \cdot t\_0, \mathsf{fma}\left(-0.16666666666666666, t\_0, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) x)))
   (if (<= y 50000000000.0)
     (fma
      (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
      (* (* y y) t_0)
      (fma -0.16666666666666666 t_0 x))
     0.0)))

C

double code(double x, double y) {
	double t_0 = (y * y) * x;
	double tmp;
	if (y <= 50000000000.0) {
		tmp = fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), ((y * y) * t_0), fma(-0.16666666666666666, t_0, x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * x)
	tmp = 0.0
	if (y <= 50000000000.0)
		tmp = fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(Float64(y * y) * t_0), fma(-0.16666666666666666, t_0, x));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, 50000000000.0], N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(-0.16666666666666666 * t$95$0 + x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if y < 5e10

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right)\right)} \]

   5. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, -0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \color{blue}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot x\right)}, \mathsf{fma}\left(-0.16666666666666666, \left(y \cdot y\right) \cdot x, x\right)\right) \]

   if 5e10 < y

   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 98.7%

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

   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 24.5

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

   7. Applied rewrites 24.5%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot x\right), \mathsf{fma}\left(-0.16666666666666666, \left(y \cdot y\right) \cdot x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 50000000000.0)
   (fma
    (* (* y y) x)
    (fma
     (* (fma -0.0001984126984126984 (* y y) 0.008333333333333333) y)
     y
     -0.16666666666666666)
    x)
   0.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 50000000000.0) {
		tmp = fma(((y * y) * x), fma((fma(-0.0001984126984126984, (y * y), 0.008333333333333333) * y), y, -0.16666666666666666), x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 50000000000.0)
		tmp = fma(Float64(Float64(y * y) * x), fma(Float64(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333) * y), y, -0.16666666666666666), x);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 50000000000.0], N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if y < 5e10

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right)\right)} \]

   5. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, -0.16666666666666666\right), x\right)} \]

   if 5e10 < y

   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 98.7%

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

   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 24.5

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

   7. Applied rewrites 24.5%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 27000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \left(x \cdot y\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 27000000000000.0) (fma y (* (* x y) -0.16666666666666666) x) 0.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 27000000000000.0) {
		tmp = fma(y, ((x * y) * -0.16666666666666666), x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 27000000000000.0)
		tmp = fma(y, Float64(Float64(x * y) * -0.16666666666666666), x);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 27000000000000.0], N[(y * N[(N[(x * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if y < 2.7e13

   1. Initial program 99.9%

      \[x \cdot \frac{\sin y}{y} \]
   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 85.9

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

   4. Applied rewrites 85.9%

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

   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 73.4

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

   7. Applied rewrites 73.4%

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

   9. Applied rewrites 73.4%

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

   if 2.7e13 < y

   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 98.7%

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

   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 24.8

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

   7. Applied rewrites 24.8%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 27000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \left(x \cdot y\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 16.1% 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 99.9%

   \[x \cdot \frac{\sin y}{y} \]
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 47.7%

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

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 14.0

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

7. Applied rewrites 14.0%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))