Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return x * (Math.sin(y) / y);
}

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (sin(y) / y);
end

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return x * (Math.sin(y) / y);
}

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (sin(y) / y);
end

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (/ (sin y) y) x))

C

double code(double x, double y) {
	return (sin(y) / y) * x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(y) / y) * x
end function

Java

public static double code(double x, double y) {
	return (Math.sin(y) / y) * x;
}

Python

def code(x, y):
	return (math.sin(y) / y) * x

Julia

function code(x, y)
	return Float64(Float64(sin(y) / y) * x)
end

MATLAB

function tmp = code(x, y)
	tmp = (sin(y) / y) * x;
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 62.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.002:\\ \;\;\;\;\frac{x}{\left(y \cdot y\right) \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, -0.16666666666666666, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sin y) y) 0.002)
   (/ x (* (* y y) 0.16666666666666666))
   (fma (* (* y y) x) -0.16666666666666666 x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 0.002) {
		tmp = x / ((y * y) * 0.16666666666666666);
	} else {
		tmp = fma(((y * y) * x), -0.16666666666666666, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.002)
		tmp = Float64(x / Float64(Float64(y * y) * 0.16666666666666666));
	else
		tmp = fma(Float64(Float64(y * y) * x), -0.16666666666666666, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.002], N[(x / N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 2e-3

   1. Initial program 99.7%

      \[x \cdot \frac{\sin y}{y} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) \cdot {y}^{2}} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}, {y}^{2}, 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\frac{7}{360} \cdot {y}^{2} + \frac{1}{6}}, {y}^{2}, 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360}, {y}^{2}, \frac{1}{6}\right)}, {y}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right)} \]

      9. lower-*.f64 23.1

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right)} \]

   7. Applied rewrites 23.1%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}} \]

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{6} \cdot {y}^{2} + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 23.5

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

   10. Applied rewrites 23.5%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 23.5%

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

   if 2e-3 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[x \cdot \frac{\sin y}{y} \]
   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. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 76.7

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

   4. Applied rewrites 76.7%

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

   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. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.002:\\ \;\;\;\;\frac{x}{\left(y \cdot y\right) \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, -0.16666666666666666, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-frac-neg2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-neg.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in y around 0

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

   1. div-sub N/A

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

   2. associate-/l* N/A

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

   3. unpow2 N/A

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

   4. associate-*l/ N/A

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

   5. *-inverses N/A

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

   6. *-lft-identity N/A

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

   7. sub-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 59.3

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

9. Applied rewrites 59.3%

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

    1. lift-/.f64 N/A

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

    2. clear-num N/A

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

    3. associate-/r/ N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-/l/ N/A

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

    7. metadata-eval N/A

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

    8. lift-neg.f64 N/A

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

    9. frac-2neg N/A

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

    10. lift-/.f64 N/A

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

    11. lower-/.f64 59.4

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

11. Applied rewrites 59.4%

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

Alternative 4: 62.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{6} \cdot {y}^{2} + 1}} \]

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 59.4

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

7. Applied rewrites 59.4%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)}} \]
8. Add Preprocessing

Alternative 5: 50.9% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 49.0%

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

6. Final simplification 49.0%

   \[\leadsto 1 \cdot x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))