Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 9.1s

Alternatives: 8

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: 59.1% 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 69.7%

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

   if 2.39999999999999991 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.8%

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

      2. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in y around 0 20.2%

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

   7. Taylor expanded in y around inf 20.2%

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

      1. *-un-lft-identity 20.2%

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

      2. *-commutative 20.2%

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

      3. times-frac 20.3%

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

      4. *-un-lft-identity 20.3%

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

      5. times-frac 20.3%

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

      6. metadata-eval 20.3%

         \[\leadsto \frac{1}{y} \cdot \left(\color{blue}{6} \cdot \frac{x}{y}\right) \]

   9. Applied egg-rr 20.3%

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

4. Final simplification 60.0%

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

Alternative 3: 65.4% accurate, 9.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return x * (1.0 / (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 / (1.0d0 + (y * (y * 0.16666666666666666d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*r/ 87.8%

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

   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. 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 65.8%

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

   1. clear-num 65.8%

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

   2. associate-/r/ 65.8%

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

   3. *-commutative 65.8%

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

   4. +-commutative 65.8%

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

   5. distribute-lft-in 65.8%

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

   6. div-inv 66.0%

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

   7. *-inverses 66.0%

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

   8. *-commutative 66.0%

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

8. Applied egg-rr 66.0%

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

9. Final simplification 66.0%

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

Alternative 4: 58.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{y}{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.1) x (/ -1.0 (/ (/ y x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1) {
		tmp = x;
	} else {
		tmp = -1.0 / ((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 <= 3.1d0) then
        tmp = x
    else
        tmp = (-1.0d0) / ((y / x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1) {
		tmp = x;
	} else {
		tmp = -1.0 / ((y / x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.1:
		tmp = x
	else:
		tmp = -1.0 / ((y / x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1)
		tmp = x;
	else
		tmp = Float64(-1.0 / Float64(Float64(y / x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1)
		tmp = x;
	else
		tmp = -1.0 / ((y / x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1], x, N[(-1.0 / N[(N[(y / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000009

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 69.7%

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

   if 3.10000000000000009 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.5%

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

      3. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 14.8%

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

      1. clear-num 18.8%

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

      2. un-div-inv 18.8%

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

   6. Applied egg-rr 18.8%

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

      1. frac-2neg 18.8%

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

      2. div-inv 18.8%

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

      3. distribute-neg-frac 18.8%

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

   8. Applied egg-rr 18.8%

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

      1. un-div-inv 18.8%

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

      2. neg-mul-1 18.8%

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

      3. associate-/l* 18.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 21.4%

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

      6. sqr-neg 21.4%

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

      7. sqrt-unprod 20.4%

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

      8. add-sqr-sqrt 20.4%

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

   10. Applied egg-rr 20.4%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{y}{x}}{y}}\\ \end{array} \]

Alternative 5: 65.4% accurate, 11.7× speedup

Math

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

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(y * Float64(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[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*r/ 87.8%

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

   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. 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 65.8%

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

   1. *-commutative 65.8%

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

   2. distribute-lft-in 65.9%

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

   3. *-commutative 65.9%

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

   4. div-inv 66.0%

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

   5. *-inverses 66.0%

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

8. Applied egg-rr 66.0%

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

9. Final simplification 66.0%

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

Alternative 6: 58.4% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5e-27) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e-27) {
		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 <= 5d-27) 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 <= 5e-27) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e-27:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e-27)
		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 <= 5e-27)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e-27], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e-27

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 69.2%

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

   if 5.0000000000000002e-27 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.6%

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

      3. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 20.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 7: 58.8% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1e-28) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e-28) {
		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 <= 1d-28) 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 <= 1e-28) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e-28:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e-28)
		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 <= 1e-28)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e-28], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999971e-29

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 69.2%

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

   if 9.99999999999999971e-29 < y

   1. Initial program 99.7%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.6%

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

      3. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 20.6%

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

      1. clear-num 24.2%

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

      2. un-div-inv 24.2%

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

   6. Applied egg-rr 24.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]

Alternative 8: 52.7% 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 56.8%

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

3. Final simplification 56.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023308 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))