Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

Alternatives: 9

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. associate-*r/ 87.1%

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

   2. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

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: 58.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(6 \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4) x (* (/ 1.0 y) (* 6.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (6.0 * (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 <= 2.4d0) then
        tmp = x
    else
        tmp = (1.0d0 / y) * (6.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (6.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4:
		tmp = x
	else:
		tmp = (1.0 / y) * (6.0 * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4)
		tmp = x;
	else
		tmp = Float64(Float64(1.0 / y) * Float64(6.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x;
	else
		tmp = (1.0 / y) * (6.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(N[(1.0 / y), $MachinePrecision] * N[(6.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 81.8%

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

   if 2.39999999999999991 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.6%

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

      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. *-un-lft-identity 99.7%

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

      2. div-inv 99.5%

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

      3. times-frac 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in y around 0 26.9%

      \[\leadsto \frac{1}{y} \cdot \frac{x}{\color{blue}{0.16666666666666666 \cdot y + \frac{1}{y}}} \]

   7. Taylor expanded in y around inf 26.9%

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\left(6 \cdot \frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(6 \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 4: 57.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.6e+71) x (* y (/ 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.6e+71) {
		tmp = x;
	} else {
		tmp = y * (1.0 / (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 <= 5.6d+71) then
        tmp = x
    else
        tmp = y * (1.0d0 / (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.6e+71) {
		tmp = x;
	} else {
		tmp = y * (1.0 / (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.6e+71:
		tmp = x
	else:
		tmp = y * (1.0 / (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.6e+71)
		tmp = x;
	else
		tmp = Float64(y * Float64(1.0 / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.6e+71)
		tmp = x;
	else
		tmp = y * (1.0 / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.6e+71], x, N[(y * N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.60000000000000004e71

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 74.0%

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

   if 5.60000000000000004e71 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l/ 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 3.6%

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

      1. *-commutative 3.6%

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

   6. Simplified 3.6%

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

      1. associate-/l* 33.6%

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

      2. div-inv 33.6%

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

   8. Applied egg-rr 33.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{y}{x}}\\ \end{array} \]

Alternative 5: 58.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4) x (/ x (* y (* y 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = x / (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d0) then
        tmp = x
    else
        tmp = x / (y * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x;
	} else {
		tmp = x / (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4:
		tmp = x
	else:
		tmp = x / (y * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4)
		tmp = x;
	else
		tmp = Float64(x / Float64(y * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x;
	else
		tmp = x / (y * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(x / N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 81.8%

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

   if 2.39999999999999991 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.6%

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

      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. clear-num 99.6%

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

      2. associate-/r/ 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in y around 0 26.9%

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

   7. Taylor expanded in y around inf 26.9%

      \[\leadsto \frac{x}{\color{blue}{\left(0.16666666666666666 \cdot y\right)} \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 26.9%

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

   9. Simplified 26.9%

      \[\leadsto \frac{x}{\color{blue}{\left(y \cdot 0.16666666666666666\right)} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \]

Alternative 6: 64.3% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*r/ 87.1%

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

   2. associate-/l* 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.8%

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

   2. associate-/r/ 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in y around 0 67.5%

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

   1. *-commutative 67.5%

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

   2. distribute-rgt-in 67.5%

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

   3. *-commutative 67.5%

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

   4. *-commutative 67.5%

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

   5. rgt-mult-inverse 67.6%

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

8. Applied egg-rr 67.6%

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

9. Final simplification 67.6%

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

Alternative 7: 57.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 6e+71) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6e+71) {
		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 <= 6d+71) 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 <= 6e+71) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6e+71:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6e+71)
		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 <= 6e+71)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6e+71], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.00000000000000025e71

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 74.0%

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

   if 6.00000000000000025e71 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l/ 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 3.6%

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

      1. *-commutative 3.6%

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

   6. Simplified 3.6%

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

      1. clear-num 3.6%

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

      2. associate-/r/ 3.6%

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

      3. *-commutative 3.6%

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

      4. associate-*r* 33.6%

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

      5. associate-*l/ 33.6%

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

      6. *-un-lft-identity 33.6%

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

   8. Applied egg-rr 33.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 8: 57.6% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.6e+71) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.6e+71) {
		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 <= 5.6d+71) 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 <= 5.6e+71) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.6e+71:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.6e+71)
		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 <= 5.6e+71)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.6e+71], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.60000000000000004e71

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 74.0%

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

   if 5.60000000000000004e71 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.6%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. div-inv 99.5%

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

      3. times-frac 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in y around 0 3.6%

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

      1. frac-times 3.7%

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

      2. *-un-lft-identity 3.7%

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

      3. un-div-inv 3.7%

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

      4. associate-/l* 3.6%

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

      5. *-commutative 3.6%

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

      6. 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 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]

Alternative 9: 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. Taylor expanded in y around 0 58.4%

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

3. Final simplification 58.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))