Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.3s

Alternatives: 10

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. 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%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

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. Final simplification 99.8%

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

Alternative 3: 62.9% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{\frac{y \cdot y}{6}}\\ \mathbf{elif}\;y \leq 6.4:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.85e+29)
   (/ x (/ (* y y) 6.0))
   (if (<= y 6.4)
     (* x (+ 1.0 (* (* y y) -0.16666666666666666)))
     (* 6.0 (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.85e+29) {
		tmp = x / ((y * y) / 6.0);
	} else if (y <= 6.4) {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 * ((x / y) / 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 <= (-2.85d+29)) then
        tmp = x / ((y * y) / 6.0d0)
    else if (y <= 6.4d0) then
        tmp = x * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = 6.0d0 * ((x / y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.85e+29) {
		tmp = x / ((y * y) / 6.0);
	} else if (y <= 6.4) {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 * ((x / y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.85e+29:
		tmp = x / ((y * y) / 6.0)
	elif y <= 6.4:
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = 6.0 * ((x / y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.85e+29)
		tmp = Float64(x / Float64(Float64(y * y) / 6.0));
	elseif (y <= 6.4)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(6.0 * Float64(Float64(x / y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.85e+29)
		tmp = x / ((y * y) / 6.0);
	elseif (y <= 6.4)
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = 6.0 * ((x / y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.85e+29], N[(x / N[(N[(y * y), $MachinePrecision] / 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.4], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.85e29

   1. Initial program 99.6%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around 0 27.0%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 27.0%

         \[\leadsto \frac{x}{1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

      2. associate-*r* 27.0%

         \[\leadsto \frac{x}{1 + \color{blue}{\left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   6. Simplified 27.0%

      \[\leadsto \frac{x}{\color{blue}{1 + \left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   7. Taylor expanded in y around inf 27.0%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 27.0%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* 27.0%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{y}}{y}} \]

   9. Simplified 27.0%

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

       1. associate-/l/ 27.0%

          \[\leadsto 6 \cdot \color{blue}{\frac{x}{y \cdot y}} \]

       2. associate-*r/ 27.0%

          \[\leadsto \color{blue}{\frac{6 \cdot x}{y \cdot y}} \]

       3. *-commutative 27.0%

          \[\leadsto \frac{\color{blue}{x \cdot 6}}{y \cdot y} \]

       4. associate-/l* 27.0%

          \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot y}{6}}} \]

   11. Applied egg-rr 27.0%

       \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot y}{6}}} \]

   if -2.85e29 < y < 6.4000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 94.5%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 94.5%

         \[\leadsto x \cdot \left(1 + -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   4. Simplified 94.5%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   if 6.4000000000000004 < y

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 23.5%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 23.5%

         \[\leadsto \frac{x}{1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

      2. associate-*r* 23.5%

         \[\leadsto \frac{x}{1 + \color{blue}{\left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   6. Simplified 23.5%

      \[\leadsto \frac{x}{\color{blue}{1 + \left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   7. Taylor expanded in y around inf 23.5%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 23.5%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* 23.5%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{y}}{y}} \]

   9. Simplified 23.5%

      \[\leadsto \color{blue}{6 \cdot \frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{\frac{y \cdot y}{6}}\\ \mathbf{elif}\;y \leq 6.4:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 4: 62.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.45\right):\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.5) (not (<= y 2.45))) (* 6.0 (/ (/ x y) y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.45)) {
		tmp = 6.0 * ((x / y) / y);
	} else {
		tmp = 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 <= (-2.5d0)) .or. (.not. (y <= 2.45d0))) then
        tmp = 6.0d0 * ((x / y) / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.45)) {
		tmp = 6.0 * ((x / y) / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.5) or not (y <= 2.45):
		tmp = 6.0 * ((x / y) / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.5) || !(y <= 2.45))
		tmp = Float64(6.0 * Float64(Float64(x / y) / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.5) || ~((y <= 2.45)))
		tmp = 6.0 * ((x / y) / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.5], N[Not[LessEqual[y, 2.45]], $MachinePrecision]], N[(6.0 * N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.5 or 2.4500000000000002 < y

   1. Initial program 99.6%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 24.0%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 24.0%

         \[\leadsto \frac{x}{1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

      2. associate-*r* 24.0%

         \[\leadsto \frac{x}{1 + \color{blue}{\left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   6. Simplified 24.0%

      \[\leadsto \frac{x}{\color{blue}{1 + \left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   7. Taylor expanded in y around inf 24.0%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 24.0%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* 24.1%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{y}}{y}} \]

   9. Simplified 24.1%

      \[\leadsto \color{blue}{6 \cdot \frac{\frac{x}{y}}{y}} \]

   if -2.5 < y < 2.4500000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.8%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.45\right):\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 62.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;\frac{6}{y \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5)
   (/ 6.0 (* y (/ y x)))
   (if (<= y 2.45) x (* 6.0 (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5) {
		tmp = 6.0 / (y * (y / x));
	} else if (y <= 2.45) {
		tmp = x;
	} else {
		tmp = 6.0 * ((x / y) / 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 <= (-2.5d0)) then
        tmp = 6.0d0 / (y * (y / x))
    else if (y <= 2.45d0) then
        tmp = x
    else
        tmp = 6.0d0 * ((x / y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5) {
		tmp = 6.0 / (y * (y / x));
	} else if (y <= 2.45) {
		tmp = x;
	} else {
		tmp = 6.0 * ((x / y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5:
		tmp = 6.0 / (y * (y / x))
	elif y <= 2.45:
		tmp = x
	else:
		tmp = 6.0 * ((x / y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5)
		tmp = Float64(6.0 / Float64(y * Float64(y / x)));
	elseif (y <= 2.45)
		tmp = x;
	else
		tmp = Float64(6.0 * Float64(Float64(x / y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5)
		tmp = 6.0 / (y * (y / x));
	elseif (y <= 2.45)
		tmp = x;
	else
		tmp = 6.0 * ((x / y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5], N[(6.0 / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.45], x, N[(6.0 * N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5

   1. Initial program 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around 0 24.5%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 24.5%

         \[\leadsto \frac{x}{1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

      2. associate-*r* 24.5%

         \[\leadsto \frac{x}{1 + \color{blue}{\left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   6. Simplified 24.5%

      \[\leadsto \frac{x}{\color{blue}{1 + \left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   7. Taylor expanded in y around inf 24.5%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 24.5%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* 24.5%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{y}}{y}} \]

   9. Simplified 24.5%

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

       1. associate-*r/ 24.5%

          \[\leadsto \color{blue}{\frac{6 \cdot \frac{x}{y}}{y}} \]

       2. associate-/l* 24.5%

          \[\leadsto \color{blue}{\frac{6}{\frac{y}{\frac{x}{y}}}} \]

       3. div-inv 24.5%

          \[\leadsto \frac{6}{\color{blue}{y \cdot \frac{1}{\frac{x}{y}}}} \]

       4. clear-num 24.5%

          \[\leadsto \frac{6}{y \cdot \color{blue}{\frac{y}{x}}} \]

   11. Applied egg-rr 24.5%

       \[\leadsto \color{blue}{\frac{6}{y \cdot \frac{y}{x}}} \]

   if -2.5 < y < 2.4500000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.8%

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

   if 2.4500000000000002 < y

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 23.5%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 23.5%

         \[\leadsto \frac{x}{1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

      2. associate-*r* 23.5%

         \[\leadsto \frac{x}{1 + \color{blue}{\left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   6. Simplified 23.5%

      \[\leadsto \frac{x}{\color{blue}{1 + \left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   7. Taylor expanded in y around inf 23.5%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 23.5%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* 23.5%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{y}}{y}} \]

   9. Simplified 23.5%

      \[\leadsto \color{blue}{6 \cdot \frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;\frac{6}{y \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 6: 62.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;\frac{x}{\frac{y \cdot y}{6}}\\ \mathbf{elif}\;y \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5)
   (/ x (/ (* y y) 6.0))
   (if (<= y 2.45) x (* 6.0 (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5) {
		tmp = x / ((y * y) / 6.0);
	} else if (y <= 2.45) {
		tmp = x;
	} else {
		tmp = 6.0 * ((x / y) / 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 <= (-2.5d0)) then
        tmp = x / ((y * y) / 6.0d0)
    else if (y <= 2.45d0) then
        tmp = x
    else
        tmp = 6.0d0 * ((x / y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5) {
		tmp = x / ((y * y) / 6.0);
	} else if (y <= 2.45) {
		tmp = x;
	} else {
		tmp = 6.0 * ((x / y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5:
		tmp = x / ((y * y) / 6.0)
	elif y <= 2.45:
		tmp = x
	else:
		tmp = 6.0 * ((x / y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5)
		tmp = Float64(x / Float64(Float64(y * y) / 6.0));
	elseif (y <= 2.45)
		tmp = x;
	else
		tmp = Float64(6.0 * Float64(Float64(x / y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5)
		tmp = x / ((y * y) / 6.0);
	elseif (y <= 2.45)
		tmp = x;
	else
		tmp = 6.0 * ((x / y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5], N[(x / N[(N[(y * y), $MachinePrecision] / 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.45], x, N[(6.0 * N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5

   1. Initial program 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around 0 24.5%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 24.5%

         \[\leadsto \frac{x}{1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

      2. associate-*r* 24.5%

         \[\leadsto \frac{x}{1 + \color{blue}{\left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   6. Simplified 24.5%

      \[\leadsto \frac{x}{\color{blue}{1 + \left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   7. Taylor expanded in y around inf 24.5%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 24.5%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* 24.5%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{y}}{y}} \]

   9. Simplified 24.5%

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

       1. associate-/l/ 24.5%

          \[\leadsto 6 \cdot \color{blue}{\frac{x}{y \cdot y}} \]

       2. associate-*r/ 24.5%

          \[\leadsto \color{blue}{\frac{6 \cdot x}{y \cdot y}} \]

       3. *-commutative 24.5%

          \[\leadsto \frac{\color{blue}{x \cdot 6}}{y \cdot y} \]

       4. associate-/l* 24.5%

          \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot y}{6}}} \]

   11. Applied egg-rr 24.5%

       \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot y}{6}}} \]

   if -2.5 < y < 2.4500000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.8%

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

   if 2.4500000000000002 < y

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 23.5%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 23.5%

         \[\leadsto \frac{x}{1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

      2. associate-*r* 23.5%

         \[\leadsto \frac{x}{1 + \color{blue}{\left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   6. Simplified 23.5%

      \[\leadsto \frac{x}{\color{blue}{1 + \left(0.16666666666666666 \cdot y\right) \cdot y}} \]

   7. Taylor expanded in y around inf 23.5%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 23.5%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* 23.5%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{y}}{y}} \]

   9. Simplified 23.5%

      \[\leadsto \color{blue}{6 \cdot \frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;\frac{x}{\frac{y \cdot y}{6}}\\ \mathbf{elif}\;y \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 7: 60.8% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+99} \lor \neg \left(y \leq 5 \cdot 10^{-41}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.1e+99) (not (<= y 5e-41))) (* y (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.1e+99) || !(y <= 5e-41)) {
		tmp = y * (x / y);
	} else {
		tmp = 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 <= (-1.1d+99)) .or. (.not. (y <= 5d-41))) then
        tmp = y * (x / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.1e+99) || !(y <= 5e-41)) {
		tmp = y * (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.1e+99) or not (y <= 5e-41):
		tmp = y * (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.1e+99) || !(y <= 5e-41))
		tmp = Float64(y * Float64(x / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.1e+99) || ~((y <= 5e-41)))
		tmp = y * (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.1e+99], N[Not[LessEqual[y, 5e-41]], $MachinePrecision]], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.09999999999999989e99 or 4.9999999999999996e-41 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in y around 0 9.0%

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

      1. associate-/l* 30.5%

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

      2. div-inv 30.5%

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

      3. clear-num 28.3%

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

   5. Applied egg-rr 28.3%

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

   if -1.09999999999999989e99 < y < 4.9999999999999996e-41

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 85.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+99} \lor \neg \left(y \leq 5 \cdot 10^{-41}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 61.1% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+129} \lor \neg \left(y \leq 7 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e+129) (not (<= y 7e-73))) (/ y (/ y x)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5e+129) || !(y <= 7e-73)) {
		tmp = y / (y / x);
	} else {
		tmp = 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 <= (-5d+129)) .or. (.not. (y <= 7d-73))) then
        tmp = y / (y / x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5e+129) || !(y <= 7e-73)) {
		tmp = y / (y / x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5e+129) or not (y <= 7e-73):
		tmp = y / (y / x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5e+129) || !(y <= 7e-73))
		tmp = Float64(y / Float64(y / x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5e+129) || ~((y <= 7e-73)))
		tmp = y / (y / x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5e+129], N[Not[LessEqual[y, 7e-73]], $MachinePrecision]], N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -5.0000000000000003e129 or 6.9999999999999995e-73 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.8%

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

   3. Taylor expanded in y around 0 14.9%

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

      1. associate-/l* 36.1%

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

      2. div-inv 36.1%

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

      3. clear-num 34.0%

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

   5. Applied egg-rr 34.0%

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

      1. clear-num 36.1%

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

      2. un-div-inv 36.1%

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

   7. Applied egg-rr 36.1%

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

   if -5.0000000000000003e129 < y < 6.9999999999999995e-73

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 81.1%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+129} \lor \neg \left(y \leq 7 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 63.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ x (+ 1.0 (* 0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	return x / (1.0 + (0.16666666666666666 * (y * y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (1.0d0 + (0.16666666666666666d0 * (y * y)))
end function

Java

public static double code(double x, double y) {
	return x / (1.0 + (0.16666666666666666 * (y * y)));
}

Python

def code(x, y):
	return x / (1.0 + (0.16666666666666666 * (y * y)))

Julia

function code(x, y)
	return Float64(x / Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (1.0 + (0.16666666666666666 * (y * y)));
end

Wolfram

code[x_, y_] := N[(x / N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around 0 61.7%

   \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 61.7%

      \[\leadsto \frac{x}{1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

   2. associate-*r* 61.7%

      \[\leadsto \frac{x}{1 + \color{blue}{\left(0.16666666666666666 \cdot y\right) \cdot y}} \]

6. Simplified 61.7%

   \[\leadsto \frac{x}{\color{blue}{1 + \left(0.16666666666666666 \cdot y\right) \cdot y}} \]

7. Taylor expanded in y around 0 61.7%

   \[\leadsto \frac{x}{1 + \color{blue}{0.16666666666666666 \cdot {y}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 61.7%

      \[\leadsto \frac{x}{1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

9. Simplified 61.7%

   \[\leadsto \frac{x}{1 + \color{blue}{0.16666666666666666 \cdot \left(y \cdot y\right)}} \]

10. Final simplification 61.7%

    \[\leadsto \frac{x}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

Alternative 10: 51.0% 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 51.7%

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

3. Final simplification 51.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023192 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))