Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.6s

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. Final simplification 99.8%

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

Alternative 2: 57.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1e-5) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e-5) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1d-5) then
        tmp = x
    else
        tmp = y * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e-5) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e-5:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e-5)
		tmp = x;
	else
		tmp = Float64(y * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e-5)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e-5], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000008e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 60.3%

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

   if 1.00000000000000008e-5 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l/ 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 5.7%

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

      1. *-commutative 5.7%

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

   7. Simplified 5.7%

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

      1. clear-num 5.7%

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

      2. inv-pow 5.7%

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

      3. associate-/r* 5.7%

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

      4. pow1 5.7%

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

      5. pow1 5.7%

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

      6. pow-div 5.7%

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

      7. metadata-eval 5.7%

         \[\leadsto {\left(\frac{{y}^{\color{blue}{0}}}{x}\right)}^{-1} \]

      8. metadata-eval 5.7%

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

   9. Applied egg-rr 5.7%

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

       1. unpow-1 5.7%

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

       2. associate-/l* 5.7%

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

       3. lft-mult-inverse 5.7%

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

       4. frac-times 28.5%

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

       5. clear-num 28.5%

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

       6. /-rgt-identity 28.5%

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

       7. *-commutative 28.5%

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

   11. Applied egg-rr 28.5%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2e-22) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-22) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2d-22) then
        tmp = x
    else
        tmp = y / (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-22) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-22:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-22)
		tmp = x;
	else
		tmp = Float64(y / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-22)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-22], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 59.4%

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

   if 2.0000000000000001e-22 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l/ 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in y around 0 12.0%

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

      1. *-commutative 12.0%

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

   7. Simplified 12.0%

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

      1. associate-/l* 34.1%

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

      2. div-inv 34.1%

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

      3. inv-pow 34.1%

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

      4. div-inv 34.1%

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

      5. metadata-eval 34.1%

         \[\leadsto y \cdot {\left(y \cdot \frac{\color{blue}{{y}^{0}}}{x}\right)}^{-1} \]

      6. metadata-eval 34.1%

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

      7. pow-div 34.1%

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

      8. pow1 34.1%

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

      9. pow1 34.1%

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

      10. associate-/r* 32.9%

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

      11. pow-prod-down 32.9%

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

      12. inv-pow 32.9%

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

      13. inv-pow 32.9%

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

      14. clear-num 32.9%

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

      15. div-inv 32.9%

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

      16. *-commutative 32.9%

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

      17. associate-*l* 34.1%

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

      18. pow1 34.1%

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

      19. inv-pow 34.1%

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

      20. pow-prod-up 34.1%

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

      21. metadata-eval 34.1%

          \[\leadsto y \cdot \left(\frac{1}{y} \cdot \left(x \cdot {y}^{\color{blue}{0}}\right)\right) \]

      22. metadata-eval 34.1%

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

      23. *-commutative 34.1%

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

      24. *-un-lft-identity 34.1%

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

   9. Applied egg-rr 34.1%

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

       1. associate-*l/ 34.1%

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

       2. *-un-lft-identity 34.1%

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

       3. associate-*r/ 12.0%

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

       4. associate-/l* 34.1%

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

   11. Applied egg-rr 34.1%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.1% accurate, 105.0× 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 99.8%

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

3. Taylor expanded in y around 0 48.2%

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

4. Final simplification 48.2%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))