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

Percentage Accurate: 100.0% → 100.0%

Time: 9.9s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{2 - \left(x + y\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (- 2.0 (+ x y))))

C

double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / (2.0d0 - (x + y))
end function

Java

public static double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Python

def code(x, y):
	return (x - y) / (2.0 - (x + y))

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) / (2.0 - (x + y));
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{2 - \left(x + y\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (- 2.0 (+ x y))))

C

double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / (2.0d0 - (x + y))
end function

Java

public static double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Python

def code(x, y):
	return (x - y) / (2.0 - (x + y))

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) / (2.0 - (x + y));
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{2 - \left(x + y\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (- 2.0 (+ x y))))

C

double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / (2.0d0 - (x + y))
end function

Java

public static double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Python

def code(x, y):
	return (x - y) / (2.0 - (x + y))

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) / (2.0 - (x + y));
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{2 - \left(x + y\right)} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x - y}{2 - \left(x + y\right)} \]
4. Add Preprocessing

Alternative 2: 61.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-73}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-103}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-243}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-229}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-48}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+30}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e+35)
   1.0
   (if (<= y -3.6e-73)
     -1.0
     (if (<= y -3.9e-103)
       (* y -0.5)
       (if (<= y -1.05e-243)
         -1.0
         (if (<= y 1.55e-229)
           (* x 0.5)
           (if (<= y 7.2e-80)
             -1.0
             (if (<= y 1e-48) (* y -0.5) (if (<= y 7e+30) -1.0 1.0)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+35) {
		tmp = 1.0;
	} else if (y <= -3.6e-73) {
		tmp = -1.0;
	} else if (y <= -3.9e-103) {
		tmp = y * -0.5;
	} else if (y <= -1.05e-243) {
		tmp = -1.0;
	} else if (y <= 1.55e-229) {
		tmp = x * 0.5;
	} else if (y <= 7.2e-80) {
		tmp = -1.0;
	} else if (y <= 1e-48) {
		tmp = y * -0.5;
	} else if (y <= 7e+30) {
		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 (y <= (-2.7d+35)) then
        tmp = 1.0d0
    else if (y <= (-3.6d-73)) then
        tmp = -1.0d0
    else if (y <= (-3.9d-103)) then
        tmp = y * (-0.5d0)
    else if (y <= (-1.05d-243)) then
        tmp = -1.0d0
    else if (y <= 1.55d-229) then
        tmp = x * 0.5d0
    else if (y <= 7.2d-80) then
        tmp = -1.0d0
    else if (y <= 1d-48) then
        tmp = y * (-0.5d0)
    else if (y <= 7d+30) 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 (y <= -2.7e+35) {
		tmp = 1.0;
	} else if (y <= -3.6e-73) {
		tmp = -1.0;
	} else if (y <= -3.9e-103) {
		tmp = y * -0.5;
	} else if (y <= -1.05e-243) {
		tmp = -1.0;
	} else if (y <= 1.55e-229) {
		tmp = x * 0.5;
	} else if (y <= 7.2e-80) {
		tmp = -1.0;
	} else if (y <= 1e-48) {
		tmp = y * -0.5;
	} else if (y <= 7e+30) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.7e+35:
		tmp = 1.0
	elif y <= -3.6e-73:
		tmp = -1.0
	elif y <= -3.9e-103:
		tmp = y * -0.5
	elif y <= -1.05e-243:
		tmp = -1.0
	elif y <= 1.55e-229:
		tmp = x * 0.5
	elif y <= 7.2e-80:
		tmp = -1.0
	elif y <= 1e-48:
		tmp = y * -0.5
	elif y <= 7e+30:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.7e+35)
		tmp = 1.0;
	elseif (y <= -3.6e-73)
		tmp = -1.0;
	elseif (y <= -3.9e-103)
		tmp = Float64(y * -0.5);
	elseif (y <= -1.05e-243)
		tmp = -1.0;
	elseif (y <= 1.55e-229)
		tmp = Float64(x * 0.5);
	elseif (y <= 7.2e-80)
		tmp = -1.0;
	elseif (y <= 1e-48)
		tmp = Float64(y * -0.5);
	elseif (y <= 7e+30)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.7e+35)
		tmp = 1.0;
	elseif (y <= -3.6e-73)
		tmp = -1.0;
	elseif (y <= -3.9e-103)
		tmp = y * -0.5;
	elseif (y <= -1.05e-243)
		tmp = -1.0;
	elseif (y <= 1.55e-229)
		tmp = x * 0.5;
	elseif (y <= 7.2e-80)
		tmp = -1.0;
	elseif (y <= 1e-48)
		tmp = y * -0.5;
	elseif (y <= 7e+30)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.7e+35], 1.0, If[LessEqual[y, -3.6e-73], -1.0, If[LessEqual[y, -3.9e-103], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -1.05e-243], -1.0, If[LessEqual[y, 1.55e-229], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 7.2e-80], -1.0, If[LessEqual[y, 1e-48], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 7e+30], -1.0, 1.0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.70000000000000003e35 or 7.00000000000000042e30 < y

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 81.0%

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

   if -2.70000000000000003e35 < y < -3.5999999999999999e-73 or -3.9000000000000002e-103 < y < -1.05e-243 or 1.55e-229 < y < 7.2e-80 or 9.9999999999999997e-49 < y < 7.00000000000000042e30

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 59.3%

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

   if -3.5999999999999999e-73 < y < -3.9000000000000002e-103 or 7.2e-80 < y < 9.9999999999999997e-49

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 83.3%

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

      1. mul-1-neg 83.3%

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

      2. distribute-neg-frac 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in y around 0 83.3%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 83.3%

         \[\leadsto \color{blue}{y \cdot -0.5} \]

   8. Simplified 83.3%

      \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -1.05e-243 < y < 1.55e-229

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 93.1%

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

   4. Taylor expanded in x around 0 70.1%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 70.1%

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   6. Simplified 70.1%

      \[\leadsto \color{blue}{x \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-73}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-103}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-243}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-229}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-48}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+30}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{y}{x}\\ \mathbf{if}\;y \leq -2.12 \cdot 10^{+83}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{-104}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-243}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-230}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-49}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+33}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ y x))))
   (if (<= y -2.12e+83)
     1.0
     (if (<= y -1.12e-72)
       t_0
       (if (<= y -4.7e-104)
         (* y -0.5)
         (if (<= y -7.2e-243)
           t_0
           (if (<= y 9.5e-230)
             (* x 0.5)
             (if (<= y 2.4e-80)
               -1.0
               (if (<= y 3.5e-49)
                 (* y -0.5)
                 (if (<= y 8e+33) -1.0 1.0))))))))))

C

double code(double x, double y) {
	double t_0 = -1.0 + (y / x);
	double tmp;
	if (y <= -2.12e+83) {
		tmp = 1.0;
	} else if (y <= -1.12e-72) {
		tmp = t_0;
	} else if (y <= -4.7e-104) {
		tmp = y * -0.5;
	} else if (y <= -7.2e-243) {
		tmp = t_0;
	} else if (y <= 9.5e-230) {
		tmp = x * 0.5;
	} else if (y <= 2.4e-80) {
		tmp = -1.0;
	} else if (y <= 3.5e-49) {
		tmp = y * -0.5;
	} else if (y <= 8e+33) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + (y / x)
    if (y <= (-2.12d+83)) then
        tmp = 1.0d0
    else if (y <= (-1.12d-72)) then
        tmp = t_0
    else if (y <= (-4.7d-104)) then
        tmp = y * (-0.5d0)
    else if (y <= (-7.2d-243)) then
        tmp = t_0
    else if (y <= 9.5d-230) then
        tmp = x * 0.5d0
    else if (y <= 2.4d-80) then
        tmp = -1.0d0
    else if (y <= 3.5d-49) then
        tmp = y * (-0.5d0)
    else if (y <= 8d+33) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 + (y / x);
	double tmp;
	if (y <= -2.12e+83) {
		tmp = 1.0;
	} else if (y <= -1.12e-72) {
		tmp = t_0;
	} else if (y <= -4.7e-104) {
		tmp = y * -0.5;
	} else if (y <= -7.2e-243) {
		tmp = t_0;
	} else if (y <= 9.5e-230) {
		tmp = x * 0.5;
	} else if (y <= 2.4e-80) {
		tmp = -1.0;
	} else if (y <= 3.5e-49) {
		tmp = y * -0.5;
	} else if (y <= 8e+33) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + (y / x)
	tmp = 0
	if y <= -2.12e+83:
		tmp = 1.0
	elif y <= -1.12e-72:
		tmp = t_0
	elif y <= -4.7e-104:
		tmp = y * -0.5
	elif y <= -7.2e-243:
		tmp = t_0
	elif y <= 9.5e-230:
		tmp = x * 0.5
	elif y <= 2.4e-80:
		tmp = -1.0
	elif y <= 3.5e-49:
		tmp = y * -0.5
	elif y <= 8e+33:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(y / x))
	tmp = 0.0
	if (y <= -2.12e+83)
		tmp = 1.0;
	elseif (y <= -1.12e-72)
		tmp = t_0;
	elseif (y <= -4.7e-104)
		tmp = Float64(y * -0.5);
	elseif (y <= -7.2e-243)
		tmp = t_0;
	elseif (y <= 9.5e-230)
		tmp = Float64(x * 0.5);
	elseif (y <= 2.4e-80)
		tmp = -1.0;
	elseif (y <= 3.5e-49)
		tmp = Float64(y * -0.5);
	elseif (y <= 8e+33)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 + (y / x);
	tmp = 0.0;
	if (y <= -2.12e+83)
		tmp = 1.0;
	elseif (y <= -1.12e-72)
		tmp = t_0;
	elseif (y <= -4.7e-104)
		tmp = y * -0.5;
	elseif (y <= -7.2e-243)
		tmp = t_0;
	elseif (y <= 9.5e-230)
		tmp = x * 0.5;
	elseif (y <= 2.4e-80)
		tmp = -1.0;
	elseif (y <= 3.5e-49)
		tmp = y * -0.5;
	elseif (y <= 8e+33)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.12e+83], 1.0, If[LessEqual[y, -1.12e-72], t$95$0, If[LessEqual[y, -4.7e-104], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -7.2e-243], t$95$0, If[LessEqual[y, 9.5e-230], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 2.4e-80], -1.0, If[LessEqual[y, 3.5e-49], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 8e+33], -1.0, 1.0]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -2.11999999999999991e83 or 7.9999999999999996e33 < y

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 82.6%

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

   if -2.11999999999999991e83 < y < -1.12000000000000005e-72 or -4.7e-104 < y < -7.2000000000000003e-243

   1. Initial program 100.0%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 59.2%

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

   6. Taylor expanded in x around 0 59.3%

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

   if -1.12000000000000005e-72 < y < -4.7e-104 or 2.3999999999999999e-80 < y < 3.50000000000000006e-49

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 83.3%

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

      1. mul-1-neg 83.3%

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

      2. distribute-neg-frac 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in y around 0 83.3%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 83.3%

         \[\leadsto \color{blue}{y \cdot -0.5} \]

   8. Simplified 83.3%

      \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -7.2000000000000003e-243 < y < 9.5000000000000004e-230

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 93.1%

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

   4. Taylor expanded in x around 0 70.1%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 70.1%

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   6. Simplified 70.1%

      \[\leadsto \color{blue}{x \cdot 0.5} \]

   if 9.5000000000000004e-230 < y < 2.3999999999999999e-80 or 3.50000000000000006e-49 < y < 7.9999999999999996e33

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 57.7%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.12 \cdot 10^{+83}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-72}:\\ \;\;\;\;-1 + \frac{y}{x}\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{-104}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-243}:\\ \;\;\;\;-1 + \frac{y}{x}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-230}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-49}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+33}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2 - x}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -7.4 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-100}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-51}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+31}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- 2.0 x))))
   (if (<= y -6.5e+37)
     1.0
     (if (<= y -7.4e-72)
       t_0
       (if (<= y -1.95e-100)
         (* y -0.5)
         (if (<= y 1.25e-81)
           t_0
           (if (<= y 1.35e-51) (* y -0.5) (if (<= y 4e+31) -1.0 1.0))))))))

C

double code(double x, double y) {
	double t_0 = x / (2.0 - x);
	double tmp;
	if (y <= -6.5e+37) {
		tmp = 1.0;
	} else if (y <= -7.4e-72) {
		tmp = t_0;
	} else if (y <= -1.95e-100) {
		tmp = y * -0.5;
	} else if (y <= 1.25e-81) {
		tmp = t_0;
	} else if (y <= 1.35e-51) {
		tmp = y * -0.5;
	} else if (y <= 4e+31) {
		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) :: t_0
    real(8) :: tmp
    t_0 = x / (2.0d0 - x)
    if (y <= (-6.5d+37)) then
        tmp = 1.0d0
    else if (y <= (-7.4d-72)) then
        tmp = t_0
    else if (y <= (-1.95d-100)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.25d-81) then
        tmp = t_0
    else if (y <= 1.35d-51) then
        tmp = y * (-0.5d0)
    else if (y <= 4d+31) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (2.0 - x);
	double tmp;
	if (y <= -6.5e+37) {
		tmp = 1.0;
	} else if (y <= -7.4e-72) {
		tmp = t_0;
	} else if (y <= -1.95e-100) {
		tmp = y * -0.5;
	} else if (y <= 1.25e-81) {
		tmp = t_0;
	} else if (y <= 1.35e-51) {
		tmp = y * -0.5;
	} else if (y <= 4e+31) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (2.0 - x)
	tmp = 0
	if y <= -6.5e+37:
		tmp = 1.0
	elif y <= -7.4e-72:
		tmp = t_0
	elif y <= -1.95e-100:
		tmp = y * -0.5
	elif y <= 1.25e-81:
		tmp = t_0
	elif y <= 1.35e-51:
		tmp = y * -0.5
	elif y <= 4e+31:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(2.0 - x))
	tmp = 0.0
	if (y <= -6.5e+37)
		tmp = 1.0;
	elseif (y <= -7.4e-72)
		tmp = t_0;
	elseif (y <= -1.95e-100)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.25e-81)
		tmp = t_0;
	elseif (y <= 1.35e-51)
		tmp = Float64(y * -0.5);
	elseif (y <= 4e+31)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (2.0 - x);
	tmp = 0.0;
	if (y <= -6.5e+37)
		tmp = 1.0;
	elseif (y <= -7.4e-72)
		tmp = t_0;
	elseif (y <= -1.95e-100)
		tmp = y * -0.5;
	elseif (y <= 1.25e-81)
		tmp = t_0;
	elseif (y <= 1.35e-51)
		tmp = y * -0.5;
	elseif (y <= 4e+31)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+37], 1.0, If[LessEqual[y, -7.4e-72], t$95$0, If[LessEqual[y, -1.95e-100], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.25e-81], t$95$0, If[LessEqual[y, 1.35e-51], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 4e+31], -1.0, 1.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.4999999999999998e37 or 3.9999999999999999e31 < y

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 81.0%

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

   if -6.4999999999999998e37 < y < -7.3999999999999997e-72 or -1.94999999999999989e-100 < y < 1.24999999999999995e-81

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 82.1%

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

   if -7.3999999999999997e-72 < y < -1.94999999999999989e-100 or 1.24999999999999995e-81 < y < 1.3499999999999999e-51

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 83.3%

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

      1. mul-1-neg 83.3%

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

      2. distribute-neg-frac 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in y around 0 83.3%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 83.3%

         \[\leadsto \color{blue}{y \cdot -0.5} \]

   8. Simplified 83.3%

      \[\leadsto \color{blue}{y \cdot -0.5} \]

   if 1.3499999999999999e-51 < y < 3.9999999999999999e31

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 65.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -7.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-100}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-51}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+31}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-73} \lor \neg \left(y \leq -7 \cdot 10^{-101}\right) \land y \leq 1.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+35)
   1.0
   (if (or (<= y -2.7e-73) (and (not (<= y -7e-101)) (<= y 1.4e-83)))
     (/ x (- 2.0 x))
     (/ y (+ y -2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+35) {
		tmp = 1.0;
	} else if ((y <= -2.7e-73) || (!(y <= -7e-101) && (y <= 1.4e-83))) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.5d+35)) then
        tmp = 1.0d0
    else if ((y <= (-2.7d-73)) .or. (.not. (y <= (-7d-101))) .and. (y <= 1.4d-83)) then
        tmp = x / (2.0d0 - x)
    else
        tmp = y / (y + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+35) {
		tmp = 1.0;
	} else if ((y <= -2.7e-73) || (!(y <= -7e-101) && (y <= 1.4e-83))) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5e+35:
		tmp = 1.0
	elif (y <= -2.7e-73) or (not (y <= -7e-101) and (y <= 1.4e-83)):
		tmp = x / (2.0 - x)
	else:
		tmp = y / (y + -2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5e+35)
		tmp = 1.0;
	elseif ((y <= -2.7e-73) || (!(y <= -7e-101) && (y <= 1.4e-83)))
		tmp = Float64(x / Float64(2.0 - x));
	else
		tmp = Float64(y / Float64(y + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.5e+35)
		tmp = 1.0;
	elseif ((y <= -2.7e-73) || (~((y <= -7e-101)) && (y <= 1.4e-83)))
		tmp = x / (2.0 - x);
	else
		tmp = y / (y + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5e+35], 1.0, If[Or[LessEqual[y, -2.7e-73], And[N[Not[LessEqual[y, -7e-101]], $MachinePrecision], LessEqual[y, 1.4e-83]]], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5000000000000003e35

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 81.5%

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

   if -6.5000000000000003e35 < y < -2.69999999999999994e-73 or -6.99999999999999989e-101 < y < 1.4e-83

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 82.1%

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

   if -2.69999999999999994e-73 < y < -6.99999999999999989e-101 or 1.4e-83 < y

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.0%

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

      1. mul-1-neg 76.0%

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

      2. distribute-neg-frac 76.0%

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

   5. Simplified 76.0%

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

      1. frac-2neg 76.0%

         \[\leadsto \color{blue}{\frac{-\left(-y\right)}{-\left(2 - y\right)}} \]

      2. div-inv 75.9%

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

      3. remove-double-neg 75.9%

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

      4. sub-neg 75.9%

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

      5. distribute-neg-in 75.9%

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

      6. metadata-eval 75.9%

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

      7. remove-double-neg 75.9%

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

   7. Applied egg-rr 75.9%

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

      1. associate-*r/ 76.0%

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

      2. *-rgt-identity 76.0%

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

      3. +-commutative 76.0%

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

   9. Simplified 76.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-73} \lor \neg \left(y \leq -7 \cdot 10^{-101}\right) \land y \leq 1.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{+35}:\\ \;\;\;\;\frac{x - y}{-y}\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-73} \lor \neg \left(y \leq -1.95 \cdot 10^{-100}\right) \land y \leq 8.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.08e+35)
   (/ (- x y) (- y))
   (if (or (<= y -2.2e-73) (and (not (<= y -1.95e-100)) (<= y 8.8e-82)))
     (/ x (- 2.0 x))
     (/ y (+ y -2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.08e+35) {
		tmp = (x - y) / -y;
	} else if ((y <= -2.2e-73) || (!(y <= -1.95e-100) && (y <= 8.8e-82))) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.08d+35)) then
        tmp = (x - y) / -y
    else if ((y <= (-2.2d-73)) .or. (.not. (y <= (-1.95d-100))) .and. (y <= 8.8d-82)) then
        tmp = x / (2.0d0 - x)
    else
        tmp = y / (y + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.08e+35) {
		tmp = (x - y) / -y;
	} else if ((y <= -2.2e-73) || (!(y <= -1.95e-100) && (y <= 8.8e-82))) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.08e+35:
		tmp = (x - y) / -y
	elif (y <= -2.2e-73) or (not (y <= -1.95e-100) and (y <= 8.8e-82)):
		tmp = x / (2.0 - x)
	else:
		tmp = y / (y + -2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.08e+35)
		tmp = Float64(Float64(x - y) / Float64(-y));
	elseif ((y <= -2.2e-73) || (!(y <= -1.95e-100) && (y <= 8.8e-82)))
		tmp = Float64(x / Float64(2.0 - x));
	else
		tmp = Float64(y / Float64(y + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.08e+35)
		tmp = (x - y) / -y;
	elseif ((y <= -2.2e-73) || (~((y <= -1.95e-100)) && (y <= 8.8e-82)))
		tmp = x / (2.0 - x);
	else
		tmp = y / (y + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.08e+35], N[(N[(x - y), $MachinePrecision] / (-y)), $MachinePrecision], If[Or[LessEqual[y, -2.2e-73], And[N[Not[LessEqual[y, -1.95e-100]], $MachinePrecision], LessEqual[y, 8.8e-82]]], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.08e35

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf 81.8%

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

      1. expm1-log1p-u 81.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{y} \cdot \left(x - y\right)\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{y} \cdot \left(x - y\right)\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x - y\right) \cdot \frac{-1}{y}}\right)} - 1 \]

      4. frac-2neg 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x - y\right) \cdot \color{blue}{\frac{--1}{-y}}\right)} - 1 \]

      5. metadata-eval 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x - y\right) \cdot \frac{\color{blue}{1}}{-y}\right)} - 1 \]

      6. un-div-inv 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x - y}{-y}}\right)} - 1 \]

   7. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x - y}{-y}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 81.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{-y}\right)\right)} \]

      2. expm1-log1p 82.0%

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

   9. Simplified 82.0%

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

   if -1.08e35 < y < -2.2e-73 or -1.94999999999999989e-100 < y < 8.79999999999999943e-82

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 82.1%

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

   if -2.2e-73 < y < -1.94999999999999989e-100 or 8.79999999999999943e-82 < y

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.0%

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

      1. mul-1-neg 76.0%

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

      2. distribute-neg-frac 76.0%

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

   5. Simplified 76.0%

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

      1. frac-2neg 76.0%

         \[\leadsto \color{blue}{\frac{-\left(-y\right)}{-\left(2 - y\right)}} \]

      2. div-inv 75.9%

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

      3. remove-double-neg 75.9%

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

      4. sub-neg 75.9%

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

      5. distribute-neg-in 75.9%

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

      6. metadata-eval 75.9%

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

      7. remove-double-neg 75.9%

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

   7. Applied egg-rr 75.9%

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

      1. associate-*r/ 76.0%

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

      2. *-rgt-identity 76.0%

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

      3. +-commutative 76.0%

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

   9. Simplified 76.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{+35}:\\ \;\;\;\;\frac{x - y}{-y}\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-73} \lor \neg \left(y \leq -1.95 \cdot 10^{-100}\right) \land y \leq 8.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+35}:\\ \;\;\;\;1 - \frac{x \cdot 2}{y}\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-73} \lor \neg \left(y \leq -1.9 \cdot 10^{-100}\right) \land y \leq 7.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e+35)
   (- 1.0 (/ (* x 2.0) y))
   (if (or (<= y -2.2e-73) (and (not (<= y -1.9e-100)) (<= y 7.2e-80)))
     (/ x (- 2.0 x))
     (/ y (+ y -2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+35) {
		tmp = 1.0 - ((x * 2.0) / y);
	} else if ((y <= -2.2e-73) || (!(y <= -1.9e-100) && (y <= 7.2e-80))) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.6d+35)) then
        tmp = 1.0d0 - ((x * 2.0d0) / y)
    else if ((y <= (-2.2d-73)) .or. (.not. (y <= (-1.9d-100))) .and. (y <= 7.2d-80)) then
        tmp = x / (2.0d0 - x)
    else
        tmp = y / (y + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+35) {
		tmp = 1.0 - ((x * 2.0) / y);
	} else if ((y <= -2.2e-73) || (!(y <= -1.9e-100) && (y <= 7.2e-80))) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.6e+35:
		tmp = 1.0 - ((x * 2.0) / y)
	elif (y <= -2.2e-73) or (not (y <= -1.9e-100) and (y <= 7.2e-80)):
		tmp = x / (2.0 - x)
	else:
		tmp = y / (y + -2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e+35)
		tmp = Float64(1.0 - Float64(Float64(x * 2.0) / y));
	elseif ((y <= -2.2e-73) || (!(y <= -1.9e-100) && (y <= 7.2e-80)))
		tmp = Float64(x / Float64(2.0 - x));
	else
		tmp = Float64(y / Float64(y + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.6e+35)
		tmp = 1.0 - ((x * 2.0) / y);
	elseif ((y <= -2.2e-73) || (~((y <= -1.9e-100)) && (y <= 7.2e-80)))
		tmp = x / (2.0 - x);
	else
		tmp = y / (y + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e+35], N[(1.0 - N[(N[(x * 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -2.2e-73], And[N[Not[LessEqual[y, -1.9e-100]], $MachinePrecision], LessEqual[y, 7.2e-80]]], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.60000000000000007e35

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 82.0%

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

      1. sub-neg 82.0%

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

      2. mul-1-neg 82.0%

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

      3. unsub-neg 82.0%

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

      4. mul-1-neg 82.0%

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

      5. remove-double-neg 82.0%

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

   5. Simplified 82.0%

      \[\leadsto \color{blue}{\left(1 - \frac{x}{y}\right) + \frac{2 - x}{y}} \]
   6. Step-by-step derivation

      1. associate-+l- 82.0%

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

      2. sub-div 82.0%

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

   7. Applied egg-rr 82.0%

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

   8. Taylor expanded in x around inf 82.0%

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

   if -2.60000000000000007e35 < y < -2.2e-73 or -1.89999999999999999e-100 < y < 7.2e-80

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 82.1%

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

   if -2.2e-73 < y < -1.89999999999999999e-100 or 7.2e-80 < y

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.0%

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

      1. mul-1-neg 76.0%

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

      2. distribute-neg-frac 76.0%

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

   5. Simplified 76.0%

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

      1. frac-2neg 76.0%

         \[\leadsto \color{blue}{\frac{-\left(-y\right)}{-\left(2 - y\right)}} \]

      2. div-inv 75.9%

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

      3. remove-double-neg 75.9%

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

      4. sub-neg 75.9%

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

      5. distribute-neg-in 75.9%

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

      6. metadata-eval 75.9%

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

      7. remove-double-neg 75.9%

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

   7. Applied egg-rr 75.9%

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

      1. associate-*r/ 76.0%

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

      2. *-rgt-identity 76.0%

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

      3. +-commutative 76.0%

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

   9. Simplified 76.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+35}:\\ \;\;\;\;1 - \frac{x \cdot 2}{y}\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-73} \lor \neg \left(y \leq -1.9 \cdot 10^{-100}\right) \land y \leq 7.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-243}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-229}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+27}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e+37)
   1.0
   (if (<= y -1.1e-243)
     -1.0
     (if (<= y 1.5e-229) (* x 0.5) (if (<= y 5e+27) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e+37) {
		tmp = 1.0;
	} else if (y <= -1.1e-243) {
		tmp = -1.0;
	} else if (y <= 1.5e-229) {
		tmp = x * 0.5;
	} else if (y <= 5e+27) {
		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 (y <= (-5d+37)) then
        tmp = 1.0d0
    else if (y <= (-1.1d-243)) then
        tmp = -1.0d0
    else if (y <= 1.5d-229) then
        tmp = x * 0.5d0
    else if (y <= 5d+27) 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 (y <= -5e+37) {
		tmp = 1.0;
	} else if (y <= -1.1e-243) {
		tmp = -1.0;
	} else if (y <= 1.5e-229) {
		tmp = x * 0.5;
	} else if (y <= 5e+27) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e+37:
		tmp = 1.0
	elif y <= -1.1e-243:
		tmp = -1.0
	elif y <= 1.5e-229:
		tmp = x * 0.5
	elif y <= 5e+27:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e+37)
		tmp = 1.0;
	elseif (y <= -1.1e-243)
		tmp = -1.0;
	elseif (y <= 1.5e-229)
		tmp = Float64(x * 0.5);
	elseif (y <= 5e+27)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e+37)
		tmp = 1.0;
	elseif (y <= -1.1e-243)
		tmp = -1.0;
	elseif (y <= 1.5e-229)
		tmp = x * 0.5;
	elseif (y <= 5e+27)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e+37], 1.0, If[LessEqual[y, -1.1e-243], -1.0, If[LessEqual[y, 1.5e-229], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 5e+27], -1.0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.99999999999999989e37 or 4.99999999999999979e27 < y

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 81.0%

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

   if -4.99999999999999989e37 < y < -1.1e-243 or 1.50000000000000001e-229 < y < 4.99999999999999979e27

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 53.3%

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

   if -1.1e-243 < y < 1.50000000000000001e-229

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 93.1%

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

   4. Taylor expanded in x around 0 70.1%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 70.1%

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   6. Simplified 70.1%

      \[\leadsto \color{blue}{x \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-243}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-229}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+27}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+30}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.3e+35) 1.0 (if (<= y 1.75e+30) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.3e+35) {
		tmp = 1.0;
	} else if (y <= 1.75e+30) {
		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 (y <= (-2.3d+35)) then
        tmp = 1.0d0
    else if (y <= 1.75d+30) 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 (y <= -2.3e+35) {
		tmp = 1.0;
	} else if (y <= 1.75e+30) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.3e+35:
		tmp = 1.0
	elif y <= 1.75e+30:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.3e+35)
		tmp = 1.0;
	elseif (y <= 1.75e+30)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.3e+35)
		tmp = 1.0;
	elseif (y <= 1.75e+30)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.3e+35], 1.0, If[LessEqual[y, 1.75e+30], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2999999999999998e35 or 1.75000000000000011e30 < y

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 81.0%

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

   if -2.2999999999999998e35 < y < 1.75000000000000011e30

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 47.5%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+30}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.4% accurate, 9.0× 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 100.0%

   \[\frac{x - y}{2 - \left(x + y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 32.8%

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

4. Final simplification 32.8%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - \left(x + y\right)\\ \frac{x}{t_0} - \frac{y}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 2.0 (+ x y)))) (- (/ x t_0) (/ y t_0))))

C

double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	return (x / t_0) - (y / t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = 2.0d0 - (x + y)
    code = (x / t_0) - (y / t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	return (x / t_0) - (y / t_0);
}

Python

def code(x, y):
	t_0 = 2.0 - (x + y)
	return (x / t_0) - (y / t_0)

Julia

function code(x, y)
	t_0 = Float64(2.0 - Float64(x + y))
	return Float64(Float64(x / t_0) - Float64(y / t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = 2.0 - (x + y);
	tmp = (x / t_0) - (y / t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]}, N[(N[(x / t$95$0), $MachinePrecision] - N[(y / t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024021 
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, C"
  :precision binary64

  :herbie-target
  (- (/ x (- 2.0 (+ x y))) (/ y (- 2.0 (+ x y))))

  (/ (- x y) (- 2.0 (+ x y))))