Data.Colour.RGB:hslsv from colour-2.3.3, D

Percentage Accurate: 100.0% → 100.0%

Time: 8.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{x + y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (+ x y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{x + y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (+ x y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{x + y}{x - y}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\frac{x - y}{x + y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x + y}{x - y}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + y\right), \color{blue}{\left(x - y\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{x} - y\right)\right)\right) \]

   5. --lowering--.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{x - y}}} \]
5. Add Preprocessing

Alternative 2: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y \cdot -2}{x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;-1 - -2 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x (* y -2.0)) x)))
   (if (<= x -5e-85) t_0 (if (<= x 3.9e+121) (- -1.0 (* -2.0 (/ x y))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + (y * -2.0)) / x;
	double tmp;
	if (x <= -5e-85) {
		tmp = t_0;
	} else if (x <= 3.9e+121) {
		tmp = -1.0 - (-2.0 * (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + (y * (-2.0d0))) / x
    if (x <= (-5d-85)) then
        tmp = t_0
    else if (x <= 3.9d+121) then
        tmp = (-1.0d0) - ((-2.0d0) * (x / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + (y * -2.0)) / x;
	double tmp;
	if (x <= -5e-85) {
		tmp = t_0;
	} else if (x <= 3.9e+121) {
		tmp = -1.0 - (-2.0 * (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + (y * -2.0)) / x
	tmp = 0
	if x <= -5e-85:
		tmp = t_0
	elif x <= 3.9e+121:
		tmp = -1.0 - (-2.0 * (x / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + Float64(y * -2.0)) / x)
	tmp = 0.0
	if (x <= -5e-85)
		tmp = t_0;
	elseif (x <= 3.9e+121)
		tmp = Float64(-1.0 - Float64(-2.0 * Float64(x / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + (y * -2.0)) / x;
	tmp = 0.0;
	if (x <= -5e-85)
		tmp = t_0;
	elseif (x <= 3.9e+121)
		tmp = -1.0 - (-2.0 * (x / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + N[(y * -2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -5e-85], t$95$0, If[LessEqual[x, 3.9e+121], N[(-1.0 - N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-85 or 3.89999999999999984e121 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1 \cdot y - y}{x}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot y - y\right), \color{blue}{x}\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot y - 1 \cdot y\right), x\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \left(-1 - 1\right)\right), x\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot -2\right), x\right)\right) \]

      9. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, -2\right), x\right)\right) \]

   5. Simplified 80.2%

      \[\leadsto \color{blue}{1 + \frac{y \cdot -2}{x}} \]

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + -2 \cdot y\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + y \cdot -2\right), x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + y \cdot \left(-1 + -1\right)\right), x\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + \left(-1 \cdot y + -1 \cdot y\right)\right), x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + \left(-1 \cdot y + \left(\mathsf{neg}\left(y\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + \left(-1 \cdot y - y\right)\right), x\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(-1 \cdot y - y\right)\right), x\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(-1 \cdot y + \left(\mathsf{neg}\left(y\right)\right)\right)\right), x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(-1 \cdot y + -1 \cdot y\right)\right), x\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(y \cdot \left(-1 + -1\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(y \cdot -2\right)\right), x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(-2 \cdot y\right)\right), x\right) \]

      13. *-lowering-*.f64 80.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-2, y\right)\right), x\right) \]

   8. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{x + -2 \cdot y}{x}} \]

   if -5.0000000000000002e-85 < x < 3.89999999999999984e121

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-2}{-1} \cdot \frac{x}{y} + \left(\mathsf{neg}\left(1\right)\right) \]

      3. times-frac N/A

         \[\leadsto \frac{-2 \cdot x}{-1 \cdot y} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{x \cdot -2}{-1 \cdot y} + \left(\mathsf{neg}\left(1\right)\right) \]

      5. metadata-eval N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{-1 \cdot x - x}{y}\right)}\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(\frac{-1 \cdot x}{y} - \color{blue}{\frac{x}{y}}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(-1 \cdot \frac{x}{y} - \frac{\color{blue}{x}}{y}\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(-1 \cdot \frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(-1 \cdot \frac{x}{y} + -1 \cdot \color{blue}{\frac{x}{y}}\right)\right) \]

      20. distribute-rgt-out N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(\frac{x}{y} \cdot \color{blue}{\left(-1 + -1\right)}\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(\frac{x}{y} \cdot -2\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{x}{y}\right), \color{blue}{-2}\right)\right) \]

      23. /-lowering-/.f64 76.9%

          \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), -2\right)\right) \]

   5. Simplified 76.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\frac{x + y \cdot -2}{x}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;-1 - -2 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot -2}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{y \cdot -2}{x}\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+121}:\\ \;\;\;\;-1 - -2 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* y -2.0) x))))
   (if (<= x -1.65e-85) t_0 (if (<= x 4e+121) (- -1.0 (* -2.0 (/ x y))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * -2.0) / x);
	double tmp;
	if (x <= -1.65e-85) {
		tmp = t_0;
	} else if (x <= 4e+121) {
		tmp = -1.0 - (-2.0 * (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((y * (-2.0d0)) / x)
    if (x <= (-1.65d-85)) then
        tmp = t_0
    else if (x <= 4d+121) then
        tmp = (-1.0d0) - ((-2.0d0) * (x / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((y * -2.0) / x);
	double tmp;
	if (x <= -1.65e-85) {
		tmp = t_0;
	} else if (x <= 4e+121) {
		tmp = -1.0 - (-2.0 * (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((y * -2.0) / x)
	tmp = 0
	if x <= -1.65e-85:
		tmp = t_0
	elif x <= 4e+121:
		tmp = -1.0 - (-2.0 * (x / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * -2.0) / x))
	tmp = 0.0
	if (x <= -1.65e-85)
		tmp = t_0;
	elseif (x <= 4e+121)
		tmp = Float64(-1.0 - Float64(-2.0 * Float64(x / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((y * -2.0) / x);
	tmp = 0.0;
	if (x <= -1.65e-85)
		tmp = t_0;
	elseif (x <= 4e+121)
		tmp = -1.0 - (-2.0 * (x / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e-85], t$95$0, If[LessEqual[x, 4e+121], N[(-1.0 - N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.64999999999999986e-85 or 4.00000000000000015e121 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1 \cdot y - y}{x}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot y - y\right), \color{blue}{x}\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot y - 1 \cdot y\right), x\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \left(-1 - 1\right)\right), x\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot -2\right), x\right)\right) \]

      9. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, -2\right), x\right)\right) \]

   5. Simplified 80.2%

      \[\leadsto \color{blue}{1 + \frac{y \cdot -2}{x}} \]

   if -1.64999999999999986e-85 < x < 4.00000000000000015e121

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-2}{-1} \cdot \frac{x}{y} + \left(\mathsf{neg}\left(1\right)\right) \]

      3. times-frac N/A

         \[\leadsto \frac{-2 \cdot x}{-1 \cdot y} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{x \cdot -2}{-1 \cdot y} + \left(\mathsf{neg}\left(1\right)\right) \]

      5. metadata-eval N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{-1 \cdot x - x}{y}\right)}\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(\frac{-1 \cdot x}{y} - \color{blue}{\frac{x}{y}}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(-1 \cdot \frac{x}{y} - \frac{\color{blue}{x}}{y}\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(-1 \cdot \frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(-1 \cdot \frac{x}{y} + -1 \cdot \color{blue}{\frac{x}{y}}\right)\right) \]

      20. distribute-rgt-out N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(\frac{x}{y} \cdot \color{blue}{\left(-1 + -1\right)}\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \left(\frac{x}{y} \cdot -2\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{x}{y}\right), \color{blue}{-2}\right)\right) \]

      23. /-lowering-/.f64 76.9%

          \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), -2\right)\right) \]

   5. Simplified 76.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-85}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+121}:\\ \;\;\;\;-1 - -2 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{y \cdot -2}{x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* y -2.0) x))))
   (if (<= x -5e-85) t_0 (if (<= x 3.9e+121) (/ (- x y) y) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * -2.0) / x);
	double tmp;
	if (x <= -5e-85) {
		tmp = t_0;
	} else if (x <= 3.9e+121) {
		tmp = (x - y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((y * (-2.0d0)) / x)
    if (x <= (-5d-85)) then
        tmp = t_0
    else if (x <= 3.9d+121) then
        tmp = (x - y) / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((y * -2.0) / x);
	double tmp;
	if (x <= -5e-85) {
		tmp = t_0;
	} else if (x <= 3.9e+121) {
		tmp = (x - y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((y * -2.0) / x)
	tmp = 0
	if x <= -5e-85:
		tmp = t_0
	elif x <= 3.9e+121:
		tmp = (x - y) / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * -2.0) / x))
	tmp = 0.0
	if (x <= -5e-85)
		tmp = t_0;
	elseif (x <= 3.9e+121)
		tmp = Float64(Float64(x - y) / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((y * -2.0) / x);
	tmp = 0.0;
	if (x <= -5e-85)
		tmp = t_0;
	elseif (x <= 3.9e+121)
		tmp = (x - y) / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-85], t$95$0, If[LessEqual[x, 3.9e+121], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-85 or 3.89999999999999984e121 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1 \cdot y - y}{x}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot y - y\right), \color{blue}{x}\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot y - 1 \cdot y\right), x\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \left(-1 - 1\right)\right), x\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot -2\right), x\right)\right) \]

      9. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, -2\right), x\right)\right) \]

   5. Simplified 80.2%

      \[\leadsto \color{blue}{1 + \frac{y \cdot -2}{x}} \]

   if -5.0000000000000002e-85 < x < 3.89999999999999984e121

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \color{blue}{y}\right) \]
   4. Step-by-step derivation

   5. Simplified 76.3%

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

Alternative 5: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+123}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-85)
   (/ x (+ x y))
   (if (<= x 1.26e+123) (/ (- x y) y) (/ (- x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-85) {
		tmp = x / (x + y);
	} else if (x <= 1.26e+123) {
		tmp = (x - y) / y;
	} else {
		tmp = (x - 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 (x <= (-5d-85)) then
        tmp = x / (x + y)
    else if (x <= 1.26d+123) then
        tmp = (x - y) / y
    else
        tmp = (x - y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e-85) {
		tmp = x / (x + y);
	} else if (x <= 1.26e+123) {
		tmp = (x - y) / y;
	} else {
		tmp = (x - y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-85:
		tmp = x / (x + y)
	elif x <= 1.26e+123:
		tmp = (x - y) / y
	else:
		tmp = (x - y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-85)
		tmp = Float64(x / Float64(x + y));
	elseif (x <= 1.26e+123)
		tmp = Float64(Float64(x - y) / y);
	else
		tmp = Float64(Float64(x - y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-85)
		tmp = x / (x + y);
	elseif (x <= 1.26e+123)
		tmp = (x - y) / y;
	else
		tmp = (x - y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-85], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.26e+123], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.0000000000000002e-85

   1. Initial program 100.0%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 77.9%

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

   if -5.0000000000000002e-85 < x < 1.26e123

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \color{blue}{y}\right) \]
   4. Step-by-step derivation

   5. Simplified 76.3%

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

   if 1.26e123 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \color{blue}{x}\right) \]
   4. Step-by-step derivation

   5. Simplified 83.8%

      \[\leadsto \frac{x - y}{\color{blue}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e-87)
   (/ x (+ x y))
   (if (<= x 3.9e+121) (+ -1.0 (/ x y)) (/ (- x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.5e-87) {
		tmp = x / (x + y);
	} else if (x <= 3.9e+121) {
		tmp = -1.0 + (x / y);
	} else {
		tmp = (x - 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 (x <= (-1.5d-87)) then
        tmp = x / (x + y)
    else if (x <= 3.9d+121) then
        tmp = (-1.0d0) + (x / y)
    else
        tmp = (x - y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.5e-87) {
		tmp = x / (x + y);
	} else if (x <= 3.9e+121) {
		tmp = -1.0 + (x / y);
	} else {
		tmp = (x - y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.5e-87:
		tmp = x / (x + y)
	elif x <= 3.9e+121:
		tmp = -1.0 + (x / y)
	else:
		tmp = (x - y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.5e-87)
		tmp = Float64(x / Float64(x + y));
	elseif (x <= 3.9e+121)
		tmp = Float64(-1.0 + Float64(x / y));
	else
		tmp = Float64(Float64(x - y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.5e-87)
		tmp = x / (x + y);
	elseif (x <= 3.9e+121)
		tmp = -1.0 + (x / y);
	else
		tmp = (x - y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.5e-87], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e+121], N[(-1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.50000000000000008e-87

   1. Initial program 100.0%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 77.9%

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

   if -1.50000000000000008e-87 < x < 3.89999999999999984e121

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \color{blue}{y}\right) \]
   4. Step-by-step derivation

   5. Simplified 76.3%

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x}{y}\right), \color{blue}{1}\right) \]

      4. /-lowering-/.f64 76.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right) \]

   7. Applied egg-rr 76.3%

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

   if 3.89999999999999984e121 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \color{blue}{x}\right) \]
   4. Step-by-step derivation

   5. Simplified 83.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= x -5e-85) t_0 (if (<= x 3.9e+121) (+ -1.0 (/ x y)) t_0))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -5e-85) {
		tmp = t_0;
	} else if (x <= 3.9e+121) {
		tmp = -1.0 + (x / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (x <= (-5d-85)) then
        tmp = t_0
    else if (x <= 3.9d+121) then
        tmp = (-1.0d0) + (x / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -5e-85) {
		tmp = t_0;
	} else if (x <= 3.9e+121) {
		tmp = -1.0 + (x / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if x <= -5e-85:
		tmp = t_0
	elif x <= 3.9e+121:
		tmp = -1.0 + (x / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (x <= -5e-85)
		tmp = t_0;
	elseif (x <= 3.9e+121)
		tmp = Float64(-1.0 + Float64(x / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (x <= -5e-85)
		tmp = t_0;
	elseif (x <= 3.9e+121)
		tmp = -1.0 + (x / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-85], t$95$0, If[LessEqual[x, 3.9e+121], N[(-1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-85 or 3.89999999999999984e121 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 79.6%

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

   if -5.0000000000000002e-85 < x < 3.89999999999999984e121

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \color{blue}{y}\right) \]
   4. Step-by-step derivation

   5. Simplified 76.3%

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x}{y}\right), \color{blue}{1}\right) \]

      4. /-lowering-/.f64 76.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right) \]

   7. Applied egg-rr 76.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-85) 1.0 (if (<= x 3.9e+121) (+ -1.0 (/ x y)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-85) {
		tmp = 1.0;
	} else if (x <= 3.9e+121) {
		tmp = -1.0 + (x / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d-85)) then
        tmp = 1.0d0
    else if (x <= 3.9d+121) then
        tmp = (-1.0d0) + (x / y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e-85) {
		tmp = 1.0;
	} else if (x <= 3.9e+121) {
		tmp = -1.0 + (x / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-85:
		tmp = 1.0
	elif x <= 3.9e+121:
		tmp = -1.0 + (x / y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-85)
		tmp = 1.0;
	elseif (x <= 3.9e+121)
		tmp = Float64(-1.0 + Float64(x / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-85)
		tmp = 1.0;
	elseif (x <= 3.9e+121)
		tmp = -1.0 + (x / y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-85], 1.0, If[LessEqual[x, 3.9e+121], N[(-1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-85 or 3.89999999999999984e121 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 79.1%

      \[\leadsto \color{blue}{1} \]

   if -5.0000000000000002e-85 < x < 3.89999999999999984e121

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \color{blue}{y}\right) \]
   4. Step-by-step derivation

   5. Simplified 76.3%

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x}{y}\right), \color{blue}{1}\right) \]

      4. /-lowering-/.f64 76.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right) \]

   7. Applied egg-rr 76.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-36}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e-36) 1.0 (if (<= x 3.9e+121) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e-36) {
		tmp = 1.0;
	} else if (x <= 3.9e+121) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d-36)) then
        tmp = 1.0d0
    else if (x <= 3.9d+121) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2e-36) {
		tmp = 1.0;
	} else if (x <= 3.9e+121) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e-36:
		tmp = 1.0
	elif x <= 3.9e+121:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e-36)
		tmp = 1.0;
	elseif (x <= 3.9e+121)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e-36)
		tmp = 1.0;
	elseif (x <= 3.9e+121)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e-36], 1.0, If[LessEqual[x, 3.9e+121], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999999e-36 or 3.89999999999999984e121 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 81.4%

      \[\leadsto \color{blue}{1} \]

   if -1.9999999999999999e-36 < x < 3.89999999999999984e121

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 73.8%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{x + y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (+ x y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\frac{x - y}{x + y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 11: 51.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 99.9%

   \[\frac{x - y}{x + y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Simplified 46.2%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x + y} - \frac{y}{x + y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ x (+ x y)) (/ y (+ x y))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024145 
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ x (+ x y)) (/ y (+ x y))))

  (/ (- x y) (+ x y)))