Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 9.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 N/A

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

   5. 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. Add Preprocessing

Alternative 2: 63.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot -0.16666666666666666, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sin y) y) 5e-6)
   (* (/ x (* y y)) 6.0)
   (fma x (* (* y y) -0.16666666666666666) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 5e-6) {
		tmp = (x / (y * y)) * 6.0;
	} else {
		tmp = fma(x, ((y * y) * -0.16666666666666666), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 5e-6)
		tmp = Float64(Float64(x / Float64(y * y)) * 6.0);
	else
		tmp = fma(x, Float64(Float64(y * y) * -0.16666666666666666), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 5e-6], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 5.00000000000000041e-6

   1. Initial program 99.6%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 28.7

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

   7. Applied rewrites 28.7%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 28.7

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

   10. Applied rewrites 28.7%

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

       1. *-rgt-identity N/A

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

       2. lift-*.f64 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. metadata-eval 28.7

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

   12. Applied rewrites 28.7%

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

   if 5.00000000000000041e-6 < (/.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

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot -0.16666666666666666, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \frac{6}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot -0.16666666666666666, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sin y) y) 5e-6)
   (* x (/ 6.0 (* y y)))
   (fma x (* (* y y) -0.16666666666666666) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 5e-6) {
		tmp = x * (6.0 / (y * y));
	} else {
		tmp = fma(x, ((y * y) * -0.16666666666666666), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 5e-6)
		tmp = Float64(x * Float64(6.0 / Float64(y * y)));
	else
		tmp = fma(x, Float64(Float64(y * y) * -0.16666666666666666), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 5e-6], N[(x * N[(6.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 5.00000000000000041e-6

   1. Initial program 99.6%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 28.7

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

   7. Applied rewrites 28.7%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.7

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

   10. Applied rewrites 28.7%

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

   if 5.00000000000000041e-6 < (/.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

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \frac{6}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot -0.16666666666666666, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 5: 63.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.00205026455026455, 0.019444444444444445\right), 0.16666666666666666\right), 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  x
  (fma
   (* y y)
   (fma
    (* y y)
    (fma (* y y) 0.00205026455026455 0.019444444444444445)
    0.16666666666666666)
   1.0)))

C

double code(double x, double y) {
	return x / fma((y * y), fma((y * y), fma((y * y), 0.00205026455026455, 0.019444444444444445), 0.16666666666666666), 1.0);
}

Julia

function code(x, y)
	return Float64(x / fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), 0.00205026455026455, 0.019444444444444445), 0.16666666666666666), 1.0))
end

Wolfram

code[x_, y_] := N[(x / N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.00205026455026455 + 0.019444444444444445), $MachinePrecision] + 0.16666666666666666), $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-sin.f64 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 N/A

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

   5. 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

   \[\leadsto \frac{x}{\color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right)\right) + 1}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right), 1\right)}} \]

   3. unpow2 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right), 1\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right), 1\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{31}{15120} \cdot {y}^{2} + \frac{7}{360}}, \frac{1}{6}\right), 1\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{31}{15120}} + \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{31}{15120}, \frac{7}{360}\right)}, \frac{1}{6}\right), 1\right)} \]

   12. unpow2 N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{31}{15120}, \frac{7}{360}\right), \frac{1}{6}\right), 1\right)} \]

   13. lower-*.f64 65.6

       \[\leadsto \frac{x}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.00205026455026455, 0.019444444444444445\right), 0.16666666666666666\right), 1\right)} \]

7. Applied rewrites 65.6%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.00205026455026455, 0.019444444444444445\right), 0.16666666666666666\right), 1\right)}} \]
8. Add Preprocessing

Alternative 6: 63.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x / N[(y * N[(y * 0.16666666666666666), $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-sin.f64 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 N/A

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

   5. 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. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 65.5

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

7. Applied rewrites 65.5%

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

Alternative 7: 51.8% accurate, 117.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 53.8%

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

   1. *-rgt-identity 53.8

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

7. Applied rewrites 53.8%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))