Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.7s

Alternatives: 6

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. clear-num 99.8%

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

   2. un-div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \frac{x}{\frac{y}{\sin y}} \]
6. 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 56.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 9.5e-23) x (/ 1.0 (/ (/ y x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.5e-23) {
		tmp = x;
	} else {
		tmp = 1.0 / ((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 <= 9.5d-23) then
        tmp = x
    else
        tmp = 1.0d0 / ((y / x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.5e-23) {
		tmp = x;
	} else {
		tmp = 1.0 / ((y / x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.5e-23:
		tmp = x
	else:
		tmp = 1.0 / ((y / x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.5e-23)
		tmp = x;
	else
		tmp = Float64(1.0 / Float64(Float64(y / x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.5e-23)
		tmp = x;
	else
		tmp = 1.0 / ((y / x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.5e-23], x, N[(1.0 / N[(N[(y / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.50000000000000058e-23

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 73.9%

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

   if 9.50000000000000058e-23 < y

   1. Initial program 99.5%

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

      1. associate-*r/ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 9.0%

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

      1. *-commutative 9.0%

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

   7. Simplified 9.0%

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

      1. add-log-exp 20.0%

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

      2. *-un-lft-identity 20.0%

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

      3. log-prod 20.0%

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

      4. metadata-eval 20.0%

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

      5. add-log-exp 9.0%

         \[\leadsto 0 + \color{blue}{\frac{y \cdot x}{y}} \]

      6. associate-/l* 21.4%

         \[\leadsto 0 + \color{blue}{y \cdot \frac{x}{y}} \]

      7. clear-num 23.0%

         \[\leadsto 0 + y \cdot \color{blue}{\frac{1}{\frac{y}{x}}} \]

      8. un-div-inv 23.0%

         \[\leadsto 0 + \color{blue}{\frac{y}{\frac{y}{x}}} \]

   9. Applied egg-rr 23.0%

      \[\leadsto \color{blue}{0 + \frac{y}{\frac{y}{x}}} \]
   10. Step-by-step derivation

       1. frac-2neg 23.0%

          \[\leadsto 0 + \color{blue}{\frac{-y}{-\frac{y}{x}}} \]

       2. div-inv 23.0%

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

       3. distribute-neg-frac2 23.0%

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

   11. Applied egg-rr 23.0%

       \[\leadsto 0 + \color{blue}{\left(-y\right) \cdot \frac{1}{\frac{y}{-x}}} \]
   12. Step-by-step derivation

       1. un-div-inv 23.0%

          \[\leadsto 0 + \color{blue}{\frac{-y}{\frac{y}{-x}}} \]

       2. clear-num 23.0%

          \[\leadsto 0 + \color{blue}{\frac{1}{\frac{\frac{y}{-x}}{-y}}} \]

       3. add-sqr-sqrt 13.3%

          \[\leadsto 0 + \frac{1}{\frac{\frac{y}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}}{-y}} \]

       4. sqrt-unprod 18.5%

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

       5. sqr-neg 18.5%

          \[\leadsto 0 + \frac{1}{\frac{\frac{y}{\sqrt{\color{blue}{x \cdot x}}}}{-y}} \]

       6. sqrt-unprod 6.4%

          \[\leadsto 0 + \frac{1}{\frac{\frac{y}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{-y}} \]

       7. add-sqr-sqrt 18.0%

          \[\leadsto 0 + \frac{1}{\frac{\frac{y}{\color{blue}{x}}}{-y}} \]

       8. add-sqr-sqrt 0.0%

          \[\leadsto 0 + \frac{1}{\frac{\frac{y}{x}}{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}} \]

       9. sqrt-unprod 9.9%

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

       10. sqr-neg 9.9%

           \[\leadsto 0 + \frac{1}{\frac{\frac{y}{x}}{\sqrt{\color{blue}{y \cdot y}}}} \]

       11. sqrt-unprod 23.0%

           \[\leadsto 0 + \frac{1}{\frac{\frac{y}{x}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}} \]

       12. add-sqr-sqrt 23.0%

           \[\leadsto 0 + \frac{1}{\frac{\frac{y}{x}}{\color{blue}{y}}} \]

   13. Applied egg-rr 23.0%

       \[\leadsto 0 + \color{blue}{\frac{1}{\frac{\frac{y}{x}}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.5% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 20000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 20000000.0) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 20000000.0) {
		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 <= 20000000.0d0) 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 <= 20000000.0) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 20000000.0:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 20000000.0)
		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 <= 20000000.0)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 20000000.0], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 73.2%

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

   if 2e7 < y

   1. Initial program 99.5%

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

      1. associate-*r/ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 4.5%

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

      1. *-commutative 4.5%

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

   7. Simplified 4.5%

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

      1. associate-/l* 18.3%

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

      2. *-un-lft-identity 18.3%

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

      3. associate-*l/ 18.3%

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

      4. *-commutative 18.3%

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

      5. associate-*l/ 18.3%

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

      6. *-un-lft-identity 18.3%

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

   9. Applied egg-rr 18.3%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 20000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 9.5e-23) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.5e-23) {
		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 <= 9.5d-23) 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 <= 9.5e-23) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.5e-23:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.5e-23)
		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 <= 9.5e-23)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.5e-23], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.50000000000000058e-23

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 73.9%

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

   if 9.50000000000000058e-23 < y

   1. Initial program 99.5%

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

      1. associate-*r/ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 9.0%

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

      1. *-commutative 9.0%

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

   7. Simplified 9.0%

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

      1. associate-/l* 21.4%

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

      2. *-un-lft-identity 21.4%

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

      3. associate-*l/ 21.4%

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

      4. *-commutative 21.4%

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

      5. associate-*l/ 21.4%

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

      6. *-un-lft-identity 21.4%

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

   9. Applied egg-rr 21.4%

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

       1. *-commutative 21.4%

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

       2. clear-num 23.0%

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

       3. div-inv 23.0%

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

   11. Applied egg-rr 23.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% 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 58.5%

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

4. Final simplification 58.5%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))