Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 6

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 99.7%

   \[x \cdot \frac{\sin y}{y} \]
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 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

Alternative 3: 63.9% accurate, 2.6× speedup

Math

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

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(y, Float64(y * fma(Float64(y * y), fma(y, Float64(y * 0.00205026455026455), 0.019444444444444445), 0.16666666666666666)), 1.0))
end

Wolfram

code[x_, y_] := N[(x / N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.00205026455026455), $MachinePrecision] + 0.019444444444444445), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

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 99.8

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

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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

7. Applied rewrites 61.5%

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

Alternative 4: 57.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 12000:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 12000.0)
   (fma x (* (* y y) -0.16666666666666666) x)
   (/ x (* y (* y 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 12000.0) {
		tmp = fma(x, ((y * y) * -0.16666666666666666), x);
	} else {
		tmp = x / (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 12000.0)
		tmp = fma(x, Float64(Float64(y * y) * -0.16666666666666666), x);
	else
		tmp = Float64(x / Float64(y * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 12000.0], N[(x * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[(x / N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 12000

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. 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 63.0

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

   5. Applied rewrites 63.0%

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

   if 12000 < y

   1. Initial program 99.5%

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

      \[\leadsto \frac{x}{\color{blue}{1 + \frac{1}{6} \cdot {y}^{2}}} \]
   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 27.3

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

   7. Applied rewrites 27.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 27.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 12000:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% 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 99.7%

   \[x \cdot \frac{\sin y}{y} \]
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 99.8

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

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 61.5

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

7. Applied rewrites 61.5%

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

Alternative 6: 51.8% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 49.3%

   \[\leadsto x \cdot \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024220 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))