Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 4.8s

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

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

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. Add Preprocessing

Alternative 2: 56.7% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4200000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{-y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4200000.0) x (* y (/ x (- y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4200000.0) {
		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 <= 4200000.0d0) 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 <= 4200000.0) {
		tmp = x;
	} else {
		tmp = y * (x / -y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4200000.0:
		tmp = x
	else:
		tmp = y * (x / -y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4200000.0)
		tmp = x;
	else
		tmp = Float64(y * Float64(x / Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4200000.0)
		tmp = x;
	else
		tmp = y * (x / -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4200000.0], x, N[(y * N[(x / (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2e6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 70.9%

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

   if 4.2e6 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 4.4%

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

      1. *-commutative 4.4%

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

   7. Simplified 4.4%

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

      1. add-log-exp 34.2%

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

      2. *-un-lft-identity 34.2%

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

      3. log-prod 34.2%

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

      4. metadata-eval 34.2%

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

      5. add-log-exp 4.4%

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

      6. associate-/l* 33.2%

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

      7. clear-num 33.9%

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

      8. un-div-inv 33.9%

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

   9. Applied egg-rr 33.9%

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

       1. frac-2neg 33.9%

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

       2. div-inv 33.9%

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

       3. distribute-neg-frac2 33.9%

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

   11. Applied egg-rr 33.9%

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

       1. expm1-log1p-u 33.5%

          \[\leadsto 0 + \left(-y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{y}{-x}}\right)\right)} \]

       2. expm1-undefine 34.4%

          \[\leadsto 0 + \left(-y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{y}{-x}}\right)} - 1\right)} \]

       3. clear-num 34.4%

          \[\leadsto 0 + \left(-y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{\frac{y}{-x}}{1}}}\right)} - 1\right) \]

       4. clear-num 34.4%

          \[\leadsto 0 + \left(-y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{y}{-x}}}\right)} - 1\right) \]

       5. clear-num 34.4%

          \[\leadsto 0 + \left(-y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{y}}\right)} - 1\right) \]

       6. add-sqr-sqrt 15.0%

          \[\leadsto 0 + \left(-y\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{y}\right)} - 1\right) \]

       7. sqrt-unprod 34.6%

          \[\leadsto 0 + \left(-y\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{y}\right)} - 1\right) \]

       8. sqr-neg 34.6%

          \[\leadsto 0 + \left(-y\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{y}\right)} - 1\right) \]

       9. sqrt-unprod 19.8%

          \[\leadsto 0 + \left(-y\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y}\right)} - 1\right) \]

       10. add-sqr-sqrt 34.7%

           \[\leadsto 0 + \left(-y\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{y}\right)} - 1\right) \]

   13. Applied egg-rr 34.7%

       \[\leadsto 0 + \left(-y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{y}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. log1p-undefine 34.7%

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

       2. rem-exp-log 35.0%

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

       3. +-commutative 35.0%

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

       4. associate--l+ 33.7%

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

       5. metadata-eval 33.7%

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

       6. *-rgt-identity 33.7%

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

       7. associate-*r/ 33.7%

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

       8. mul0-lft 33.7%

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

       9. distribute-rgt-in 33.7%

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

       10. +-rgt-identity 33.7%

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

       11. associate-*l/ 33.7%

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

       12. *-lft-identity 33.7%

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

   15. Simplified 33.7%

       \[\leadsto 0 + \left(-y\right) \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4200000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.9% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2000000.0) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2000000.0) {
		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 <= 2000000.0d0) 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 <= 2000000.0) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2000000.0:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2000000.0)
		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 <= 2000000.0)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2000000.0], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 70.9%

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

   if 2e6 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 4.4%

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

      1. *-commutative 4.4%

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

   7. Simplified 4.4%

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

      1. associate-/l* 33.2%

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

      2. *-un-lft-identity 33.2%

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

      3. associate-*l/ 33.2%

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

      4. *-commutative 33.2%

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

      5. associate-*l/ 33.2%

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

      6. *-un-lft-identity 33.2%

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

   9. Applied egg-rr 33.2%

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

       1. *-commutative 33.2%

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

       2. clear-num 33.9%

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

       3. div-inv 33.9%

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

   11. Applied egg-rr 33.9%

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

Alternative 4: 56.6% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2000000.0) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2000000.0) {
		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 <= 2000000.0d0) 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 <= 2000000.0) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2000000.0:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2000000.0)
		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 <= 2000000.0)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2000000.0], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 70.9%

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

   if 2e6 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 4.4%

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

      1. *-commutative 4.4%

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

   7. Simplified 4.4%

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

      1. associate-/l* 33.2%

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

      2. *-un-lft-identity 33.2%

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

      3. associate-*l/ 33.2%

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

      4. *-commutative 33.2%

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

      5. associate-*l/ 33.2%

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

      6. *-un-lft-identity 33.2%

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

   9. Applied egg-rr 33.2%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.8% 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 51.4%

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

Reproduce

herbie shell --seed 2024106 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))