Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.0s

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

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

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. Final simplification 99.9%

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

Alternative 2: 57.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1800000000000:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1800000000000.0)
   (* x (+ 1.0 (* (* y y) -0.16666666666666666)))
   (/ 6.0 (* y (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1800000000000.0) {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 / (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 <= 1800000000000.0d0) then
        tmp = x * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = 6.0d0 / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1800000000000.0) {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1800000000000.0:
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = 6.0 / (y * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1800000000000.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(6.0 / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1800000000000.0)
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = 6.0 / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1800000000000.0], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8e12

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 69.3%

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

      1. unpow2 69.3%

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

   4. Simplified 69.3%

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

   if 1.8e12 < y

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   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 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 37.1%

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

      1. unpow2 37.1%

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

      2. associate-*r* 37.1%

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

   6. Simplified 37.1%

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

   7. Taylor expanded in y around inf 37.1%

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

      1. associate-*r/ 37.1%

         \[\leadsto \color{blue}{\frac{6 \cdot x}{{y}^{2}}} \]

      2. unpow2 37.1%

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

      3. times-frac 37.2%

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

   9. Simplified 37.2%

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

       1. *-commutative 37.2%

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

       2. clear-num 37.2%

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

       3. frac-times 37.2%

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

       4. metadata-eval 37.2%

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

   11. Applied egg-rr 37.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1800000000000:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 3: 57.2% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 2.4500000000000002

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.7%

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

   if 2.4500000000000002 < 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. 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 36.1%

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

      1. unpow2 36.1%

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

      2. associate-*r* 36.1%

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

   6. Simplified 36.1%

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

   7. Taylor expanded in y around inf 36.1%

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

      1. associate-*r/ 36.1%

         \[\leadsto \color{blue}{\frac{6 \cdot x}{{y}^{2}}} \]

      2. unpow2 36.1%

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

      3. times-frac 36.2%

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

   9. Simplified 36.2%

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

4. Final simplification 61.0%

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

Alternative 4: 57.2% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], x, N[(6.0 / N[(y * N[(y / x), $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 70.7%

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

   if 2.39999999999999991 < 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. 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 36.1%

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

      1. unpow2 36.1%

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

      2. associate-*r* 36.1%

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

   6. Simplified 36.1%

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

   7. Taylor expanded in y around inf 36.1%

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

      1. associate-*r/ 36.1%

         \[\leadsto \color{blue}{\frac{6 \cdot x}{{y}^{2}}} \]

      2. unpow2 36.1%

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

      3. times-frac 36.2%

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

   9. Simplified 36.2%

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

       1. *-commutative 36.2%

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

       2. clear-num 36.2%

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

       3. frac-times 36.2%

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

       4. metadata-eval 36.2%

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

   11. Applied egg-rr 36.2%

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

4. Final simplification 61.0%

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

Alternative 5: 62.8% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \frac{\sin y}{y} \]
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 66.0%

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

   1. unpow2 66.0%

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

   2. associate-*r* 66.0%

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

6. Simplified 66.0%

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

   1. *-commutative 66.0%

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

   2. add-sqr-sqrt 33.2%

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

   3. sqrt-unprod 65.8%

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

   4. *-commutative 65.8%

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

   5. *-commutative 65.8%

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

   6. swap-sqr 65.8%

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

   7. metadata-eval 65.8%

      \[\leadsto \frac{x}{1 + y \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{0.027777777777777776}}} \]

   8. metadata-eval 65.8%

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

   9. swap-sqr 65.8%

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

   10. sqrt-unprod 32.6%

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

   11. add-sqr-sqrt 65.8%

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

   12. pow1 65.8%

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

8. Applied egg-rr 65.8%

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

   1. unpow1 65.8%

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

10. Simplified 65.8%

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

11. Taylor expanded in y around 0 65.8%

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

    1. unpow2 65.8%

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

13. Simplified 65.8%

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

14. Final simplification 65.8%

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

Alternative 6: 63.1% 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 99.9%

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

   1. add-cube-cbrt 99.3%

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{\frac{y}{\sin y}} \cdot \sqrt[3]{\frac{y}{\sin y}}\right) \cdot \sqrt[3]{\frac{y}{\sin y}}}} \]

   2. pow3 99.3%

      \[\leadsto \frac{x}{\color{blue}{{\left(\sqrt[3]{\frac{y}{\sin y}}\right)}^{3}}} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{x}{\color{blue}{{\left(\sqrt[3]{\frac{y}{\sin y}}\right)}^{3}}} \]

6. Taylor expanded in y around 0 66.0%

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

   1. unpow2 66.0%

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

8. Simplified 66.0%

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

9. Final simplification 66.0%

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

Alternative 7: 56.6% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.5e-8) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-8) {
		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 <= 2.5d-8) 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 <= 2.5e-8) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.5e-8:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5e-8)
		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 <= 2.5e-8)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e-8], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4999999999999999e-8

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.7%

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

   if 2.4999999999999999e-8 < y

   1. Initial program 99.7%

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

   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 4.4%

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

      1. associate-/l* 35.3%

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

      2. div-inv 35.3%

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

      3. clear-num 33.2%

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

   5. Applied egg-rr 33.2%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 8: 57.0% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.002) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.002) {
		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 <= 0.002d0) 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 <= 0.002) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.002:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.002)
		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 <= 0.002)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.002], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e-3

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.7%

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

   if 2e-3 < y

   1. Initial program 99.7%

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

   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 4.4%

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

      1. associate-/l* 35.3%

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

      2. div-inv 35.3%

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

      3. clear-num 33.2%

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

   5. Applied egg-rr 33.2%

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

      1. clear-num 35.3%

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

      2. un-div-inv 35.3%

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

   7. Applied egg-rr 35.3%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]

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 99.9%

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

2. Taylor expanded in y around 0 52.1%

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

3. Final simplification 52.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023229 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))