Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

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 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 57.0% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-32}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot 0.16666666666666666 + \frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e-32) x (/ (/ x y) (+ (* y 0.16666666666666666) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e-32) {
		tmp = x;
	} else {
		tmp = (x / y) / ((y * 0.16666666666666666) + (1.0 / 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 <= 5d-32) then
        tmp = x
    else
        tmp = (x / y) / ((y * 0.16666666666666666d0) + (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e-32) {
		tmp = x;
	} else {
		tmp = (x / y) / ((y * 0.16666666666666666) + (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e-32:
		tmp = x
	else:
		tmp = (x / y) / ((y * 0.16666666666666666) + (1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e-32)
		tmp = x;
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(y * 0.16666666666666666) + Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e-32)
		tmp = x;
	else
		tmp = (x / y) / ((y * 0.16666666666666666) + (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e-32], x, N[(N[(x / y), $MachinePrecision] / N[(N[(y * 0.16666666666666666), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5e-32

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 60.3%

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

   if 5e-32 < y

   1. Initial program 99.7%

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

      1. clear-num 99.6%

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

      2. div-inv 99.7%

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

      3. div-inv 99.6%

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

      4. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 38.8%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{0.16666666666666666 \cdot y + \frac{1}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-32}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot 0.16666666666666666 + \frac{1}{y}}\\ \end{array} \]

Alternative 3: 56.7% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{y \cdot \frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.15e-8) x (/ (- y) (* y (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.15e-8) {
		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 <= 2.15d-8) 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 <= 2.15e-8) {
		tmp = x;
	} else {
		tmp = -y / (y * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.15e-8:
		tmp = x
	else:
		tmp = -y / (y * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.15e-8)
		tmp = x;
	else
		tmp = Float64(Float64(-y) / Float64(y * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.15e-8)
		tmp = x;
	else
		tmp = -y / (y * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.15e-8], x, N[((-y) / N[(y * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1500000000000001e-8

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 61.1%

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

   if 2.1500000000000001e-8 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.6%

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

      2. clear-num 99.7%

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

      3. *-commutative 99.7%

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

      4. div-inv 99.5%

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

      5. associate-*l* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around 0 32.6%

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

      1. associate-/r/ 33.6%

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

   8. Applied egg-rr 33.6%

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

      1. un-div-inv 33.6%

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

      2. div-inv 33.6%

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

      3. associate-/l/ 5.3%

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

      4. div-inv 5.3%

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

      5. frac-2neg 5.3%

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

      6. frac-times 33.6%

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

      7. distribute-neg-frac 33.6%

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

      8. metadata-eval 33.6%

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

   10. Applied egg-rr 33.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{y \cdot \frac{-1}{x}}\\ \end{array} \]

Alternative 4: 56.3% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4.5e-18) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-18) {
		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 <= 4.5d-18) 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 <= 4.5e-18) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e-18:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-18)
		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 <= 4.5e-18)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e-18], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.49999999999999994e-18

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 60.9%

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

   if 4.49999999999999994e-18 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around 0 33.6%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5: 56.7% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.4e-8) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-8) {
		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 <= 2.4d-8) 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 <= 2.4e-8) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4e-8:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4e-8)
		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 <= 2.4e-8)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4e-8], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999998e-8

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 61.1%

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

   if 2.39999999999999998e-8 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around 0 32.6%

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

      1. associate-*l/ 5.3%

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

      2. *-commutative 5.3%

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

      3. associate-/l* 33.6%

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

   8. Applied egg-rr 33.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]

Alternative 6: 51.1% 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. Taylor expanded in y around 0 47.0%

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

3. Final simplification 47.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))