Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.9s

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 99.8%

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

Alternative 2: 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 3: 57.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{-x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e+36)
   (fma (* (* y y) x) -0.16666666666666666 x)
   (* (- y) (/ (- x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e+36) {
		tmp = fma(((y * y) * x), -0.16666666666666666, x);
	} else {
		tmp = -y * (-x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5e+36)
		tmp = fma(Float64(Float64(y * y) * x), -0.16666666666666666, x);
	else
		tmp = Float64(Float64(-y) * Float64(Float64(-x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e+36], N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision], N[((-y) * N[((-x) / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.49999999999999988e36

   1. Initial program 99.9%

      \[x \cdot \frac{\sin y}{y} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 99.7

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

   6. Applied rewrites 99.7%

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

   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 66.6

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

   9. Applied rewrites 66.6%

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

   if 2.49999999999999988e36 < 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

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.7

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

   7. Applied rewrites 4.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. neg-mul-1 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 29.6

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

   9. Applied rewrites 29.6%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{-x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.6% 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 61.4

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

7. Applied rewrites 61.4%

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

Alternative 5: 52.1% 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 48.9%

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

6. Final simplification 48.9%

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

Reproduce

herbie shell --seed 2024294 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))