Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.8s

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

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

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: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.01:\\ \;\;\;\;\frac{x}{y} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, -0.16666666666666666, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sin y) y) 0.01)
   (* (/ x y) y)
   (fma (* (* y y) x) -0.16666666666666666 x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 0.01) {
		tmp = (x / y) * y;
	} else {
		tmp = fma(((y * y) * x), -0.16666666666666666, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.01)
		tmp = Float64(Float64(x / y) * y);
	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], 0.01], N[(N[(x / y), $MachinePrecision] * y), $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) < 0.0100000000000000002

   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. associate-*r/ N/A

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

      4. frac-2neg N/A

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

      5. neg-sub0 N/A

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

      6. flip-- N/A

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

      7. +-lft-identity N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 26.8

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

   7. Applied rewrites 26.8%

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

   if 0.0100000000000000002 < (/.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. 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 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   8. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.4

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

   9. Applied rewrites 99.4%

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

Alternative 3: 63.5% 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 64.6

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

7. Applied rewrites 64.6%

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

Alternative 4: 51.8% 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 52.5%

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

6. Final simplification 52.5%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))