Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.8s

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

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

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 99.9%

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

Alternative 2: 58.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{y}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2e+24) x (* y (/ y (* y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e+24) {
		tmp = x;
	} else {
		tmp = y * (y / (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+24) then
        tmp = x
    else
        tmp = y * (y / (y * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e+24) {
		tmp = x;
	} else {
		tmp = y * (y / (y * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e+24:
		tmp = x
	else:
		tmp = y * (y / (y * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e+24)
		tmp = x;
	else
		tmp = Float64(y * Float64(y / Float64(y * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e+24)
		tmp = x;
	else
		tmp = y * (y / (y * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e+24], x, N[(y * N[(y / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e24

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.5%

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

   if 2e24 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 30.3%

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

      1. clear-num 32.2%

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

      2. associate-/r/ 30.3%

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

   8. Applied egg-rr 30.3%

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

      1. associate-/r/ 32.2%

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

      2. lft-mult-inverse 32.2%

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

      3. *-commutative 32.2%

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

      4. associate-*l/ 32.2%

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

      5. frac-times 32.9%

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

      6. *-commutative 32.9%

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

      7. *-un-lft-identity 32.9%

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

   10. Applied egg-rr 32.9%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{y}{y \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.55e-14) x (/ y (* y (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.55e-14) {
		tmp = x;
	} else {
		tmp = y / (y * (1.0 / 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 <= 2.55d-14) then
        tmp = x
    else
        tmp = y / (y * (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.55e-14) {
		tmp = x;
	} else {
		tmp = y / (y * (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.55e-14:
		tmp = x
	else:
		tmp = y / (y * (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.55e-14)
		tmp = x;
	else
		tmp = Float64(y / Float64(y * Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.55e-14)
		tmp = x;
	else
		tmp = y / (y * (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.55e-14], x, N[(y / N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5499999999999999e-14

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.6%

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

   if 2.5499999999999999e-14 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 32.4%

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

      1. clear-num 34.2%

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

      2. un-div-inv 34.2%

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

   8. Applied egg-rr 34.2%

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

      1. clear-num 34.2%

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

      2. associate-/r/ 34.2%

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

   10. Applied egg-rr 34.2%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4e+22) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e+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 <= 4d+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 <= 4e+22) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e+22:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e+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 <= 4e+22)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e+22], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4e22

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.5%

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

   if 4e22 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 30.3%

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

      1. clear-num 32.2%

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

      2. un-div-inv 32.2%

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

   8. Applied egg-rr 32.2%

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

Alternative 5: 57.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.75 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.75e-16) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.75e-16) {
		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 <= 3.75d-16) 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 <= 3.75e-16) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.75e-16:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.75e-16)
		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 <= 3.75e-16)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.75e-16], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.75e-16

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.4%

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

   if 3.75e-16 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 33.3%

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

Alternative 6: 52.3% 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 99.9%

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

3. Taylor expanded in y around 0 48.3%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))