Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.3s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 57.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4300000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.16666666666666666 \cdot \left(y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4300000000.0)
   (* x (fma (* -0.16666666666666666 y) y 1.0))
   (/ x (* 0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4300000000.0) {
		tmp = x * fma((-0.16666666666666666 * y), y, 1.0);
	} else {
		tmp = x / (0.16666666666666666 * (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4300000000.0)
		tmp = Float64(x * fma(Float64(-0.16666666666666666 * y), y, 1.0));
	else
		tmp = Float64(x / Float64(0.16666666666666666 * Float64(y * y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4300000000.0], N[(x * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3e9

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.8

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

   5. Applied rewrites 63.8%

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

   7. Applied rewrites 63.8%

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

   if 4.3e9 < y

   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 99.6

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

   4. Applied rewrites 99.6%

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

   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 32.4

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

   7. Applied rewrites 32.4%

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

   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 32.4%

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

Alternative 3: 63.8% 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 64.3

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

7. Applied rewrites 64.3%

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

Alternative 4: 51.7% 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 99.8%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 49.2%

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

Reproduce

herbie shell --seed 2024324 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))