Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 10.3s

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

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

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. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-lowering-sin.f64 99.7

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]
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: 56.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e+15)
   (* x (fma y (* y -0.16666666666666666) 1.0))
   (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e+15) {
		tmp = x * fma(y, (y * -0.16666666666666666), 1.0);
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e+15)
		tmp = Float64(x * fma(y, Float64(y * -0.16666666666666666), 1.0));
	else
		tmp = Float64(y * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e+15], N[(x * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e15

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 59.6

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

   5. Simplified 59.6%

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

   if 1.2e15 < y

   1. Initial program 99.6%

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

      1. frac-2neg N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. +-lft-identity N/A

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

      5. associate-/r/ N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 70.8%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 4.5

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

   7. Simplified 4.5%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 17.8

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

   9. Applied egg-rr 17.8%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.4% 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. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-lowering-sin.f64 99.7

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

4. Applied egg-rr 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. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 56.3

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

7. Simplified 56.3%

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

Alternative 5: 51.2% 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 99.7%

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

3. Taylor expanded in y around 0

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

5. Simplified 46.0%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))