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

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

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, 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]]

Derivation

1. Initial program 100.0%

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

   1. associate--r+ 100.0%

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

3. Simplified 100.0%

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

   1. div-sub 100.0%

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

   2. associate--l- 100.0%

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

   3. associate--l- 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 59.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x} + -1\\ t_1 := 1 - \frac{x}{y}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-165}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-265}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ y x) -1.0)) (t_1 (- 1.0 (/ x y))))
   (if (<= x -2.4e+29)
     t_0
     (if (<= x -1.85e-9)
       t_1
       (if (<= x -6.5e-165)
         (* x 0.5)
         (if (<= x -1.7e-293)
           t_1
           (if (<= x 1.6e-265) (* y -0.5) (if (<= x 4.3e+40) 1.0 t_0))))))))

C

double code(double x, double y) {
	double t_0 = (y / x) + -1.0;
	double t_1 = 1.0 - (x / y);
	double tmp;
	if (x <= -2.4e+29) {
		tmp = t_0;
	} else if (x <= -1.85e-9) {
		tmp = t_1;
	} else if (x <= -6.5e-165) {
		tmp = x * 0.5;
	} else if (x <= -1.7e-293) {
		tmp = t_1;
	} else if (x <= 1.6e-265) {
		tmp = y * -0.5;
	} else if (x <= 4.3e+40) {
		tmp = 1.0;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (y / x) + (-1.0d0)
    t_1 = 1.0d0 - (x / y)
    if (x <= (-2.4d+29)) then
        tmp = t_0
    else if (x <= (-1.85d-9)) then
        tmp = t_1
    else if (x <= (-6.5d-165)) then
        tmp = x * 0.5d0
    else if (x <= (-1.7d-293)) then
        tmp = t_1
    else if (x <= 1.6d-265) then
        tmp = y * (-0.5d0)
    else if (x <= 4.3d+40) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y / x) + -1.0;
	double t_1 = 1.0 - (x / y);
	double tmp;
	if (x <= -2.4e+29) {
		tmp = t_0;
	} else if (x <= -1.85e-9) {
		tmp = t_1;
	} else if (x <= -6.5e-165) {
		tmp = x * 0.5;
	} else if (x <= -1.7e-293) {
		tmp = t_1;
	} else if (x <= 1.6e-265) {
		tmp = y * -0.5;
	} else if (x <= 4.3e+40) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / x) + -1.0
	t_1 = 1.0 - (x / y)
	tmp = 0
	if x <= -2.4e+29:
		tmp = t_0
	elif x <= -1.85e-9:
		tmp = t_1
	elif x <= -6.5e-165:
		tmp = x * 0.5
	elif x <= -1.7e-293:
		tmp = t_1
	elif x <= 1.6e-265:
		tmp = y * -0.5
	elif x <= 4.3e+40:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / x) + -1.0)
	t_1 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (x <= -2.4e+29)
		tmp = t_0;
	elseif (x <= -1.85e-9)
		tmp = t_1;
	elseif (x <= -6.5e-165)
		tmp = Float64(x * 0.5);
	elseif (x <= -1.7e-293)
		tmp = t_1;
	elseif (x <= 1.6e-265)
		tmp = Float64(y * -0.5);
	elseif (x <= 4.3e+40)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / x) + -1.0;
	t_1 = 1.0 - (x / y);
	tmp = 0.0;
	if (x <= -2.4e+29)
		tmp = t_0;
	elseif (x <= -1.85e-9)
		tmp = t_1;
	elseif (x <= -6.5e-165)
		tmp = x * 0.5;
	elseif (x <= -1.7e-293)
		tmp = t_1;
	elseif (x <= 1.6e-265)
		tmp = y * -0.5;
	elseif (x <= 4.3e+40)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4e+29], t$95$0, If[LessEqual[x, -1.85e-9], t$95$1, If[LessEqual[x, -6.5e-165], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, -1.7e-293], t$95$1, If[LessEqual[x, 1.6e-265], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 4.3e+40], 1.0, t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.4000000000000001e29 or 4.3000000000000002e40 < x

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.7%

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

      3. associate--l- 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around inf 81.8%

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

   7. Taylor expanded in x around 0 82.1%

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

   if -2.4000000000000001e29 < x < -1.85e-9 or -6.5000000000000004e-165 < x < -1.7e-293

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.7%

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

      3. associate--l- 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around inf 65.4%

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

   7. Taylor expanded in y around 0 65.6%

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

      1. mul-1-neg 65.6%

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

      2. unsub-neg 65.6%

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

   9. Simplified 65.6%

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

   if -1.85e-9 < x < -6.5000000000000004e-165

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 62.5%

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

   5. Taylor expanded in x around 0 60.5%

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

      1. *-commutative 60.5%

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

   7. Simplified 60.5%

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

   if -1.7e-293 < x < 1.6e-265

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 91.7%

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

      1. mul-1-neg 91.7%

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

      2. distribute-neg-frac 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in y around 0 65.9%

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

      1. *-commutative 65.9%

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

   9. Simplified 65.9%

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

   if 1.6e-265 < x < 4.3000000000000002e40

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 67.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{x} + -1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-9}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-165}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-293}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-265}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \]

Alternative 3: 59.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-160}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-308}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-265}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.6e+24)
   -1.0
   (if (<= x -2.1e-9)
     1.0
     (if (<= x -4.2e-160)
       (* x 0.5)
       (if (<= x 3.6e-308)
         1.0
         (if (<= x 1.85e-265) (* y -0.5) (if (<= x 1.4e+40) 1.0 -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.6e+24) {
		tmp = -1.0;
	} else if (x <= -2.1e-9) {
		tmp = 1.0;
	} else if (x <= -4.2e-160) {
		tmp = x * 0.5;
	} else if (x <= 3.6e-308) {
		tmp = 1.0;
	} else if (x <= 1.85e-265) {
		tmp = y * -0.5;
	} else if (x <= 1.4e+40) {
		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 <= (-7.6d+24)) then
        tmp = -1.0d0
    else if (x <= (-2.1d-9)) then
        tmp = 1.0d0
    else if (x <= (-4.2d-160)) then
        tmp = x * 0.5d0
    else if (x <= 3.6d-308) then
        tmp = 1.0d0
    else if (x <= 1.85d-265) then
        tmp = y * (-0.5d0)
    else if (x <= 1.4d+40) 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 <= -7.6e+24) {
		tmp = -1.0;
	} else if (x <= -2.1e-9) {
		tmp = 1.0;
	} else if (x <= -4.2e-160) {
		tmp = x * 0.5;
	} else if (x <= 3.6e-308) {
		tmp = 1.0;
	} else if (x <= 1.85e-265) {
		tmp = y * -0.5;
	} else if (x <= 1.4e+40) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.6e+24:
		tmp = -1.0
	elif x <= -2.1e-9:
		tmp = 1.0
	elif x <= -4.2e-160:
		tmp = x * 0.5
	elif x <= 3.6e-308:
		tmp = 1.0
	elif x <= 1.85e-265:
		tmp = y * -0.5
	elif x <= 1.4e+40:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.6e+24)
		tmp = -1.0;
	elseif (x <= -2.1e-9)
		tmp = 1.0;
	elseif (x <= -4.2e-160)
		tmp = Float64(x * 0.5);
	elseif (x <= 3.6e-308)
		tmp = 1.0;
	elseif (x <= 1.85e-265)
		tmp = Float64(y * -0.5);
	elseif (x <= 1.4e+40)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.6e+24)
		tmp = -1.0;
	elseif (x <= -2.1e-9)
		tmp = 1.0;
	elseif (x <= -4.2e-160)
		tmp = x * 0.5;
	elseif (x <= 3.6e-308)
		tmp = 1.0;
	elseif (x <= 1.85e-265)
		tmp = y * -0.5;
	elseif (x <= 1.4e+40)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.6e+24], -1.0, If[LessEqual[x, -2.1e-9], 1.0, If[LessEqual[x, -4.2e-160], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 3.6e-308], 1.0, If[LessEqual[x, 1.85e-265], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 1.4e+40], 1.0, -1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.6000000000000003e24 or 1.4000000000000001e40 < x

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 81.0%

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

   if -7.6000000000000003e24 < x < -2.10000000000000019e-9 or -4.2000000000000001e-160 < x < 3.5999999999999999e-308 or 1.8499999999999999e-265 < x < 1.4000000000000001e40

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 66.8%

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

   if -2.10000000000000019e-9 < x < -4.2000000000000001e-160

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 62.5%

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

   5. Taylor expanded in x around 0 60.5%

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

      1. *-commutative 60.5%

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

   7. Simplified 60.5%

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

   if 3.5999999999999999e-308 < x < 1.8499999999999999e-265

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 91.7%

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

      1. mul-1-neg 91.7%

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

      2. distribute-neg-frac 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in y around 0 65.9%

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

      1. *-commutative 65.9%

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

   9. Simplified 65.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-160}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-308}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-265}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 60.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-160}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-299}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-265}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= x -7.2e+28)
     -1.0
     (if (<= x -1.85e-9)
       t_0
       (if (<= x -4.2e-160)
         (* x 0.5)
         (if (<= x -1.75e-299)
           t_0
           (if (<= x 1.85e-265) (* y -0.5) (if (<= x 2e+42) 1.0 -1.0))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (x <= -7.2e+28) {
		tmp = -1.0;
	} else if (x <= -1.85e-9) {
		tmp = t_0;
	} else if (x <= -4.2e-160) {
		tmp = x * 0.5;
	} else if (x <= -1.75e-299) {
		tmp = t_0;
	} else if (x <= 1.85e-265) {
		tmp = y * -0.5;
	} else if (x <= 2e+42) {
		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 - (x / y)
    if (x <= (-7.2d+28)) then
        tmp = -1.0d0
    else if (x <= (-1.85d-9)) then
        tmp = t_0
    else if (x <= (-4.2d-160)) then
        tmp = x * 0.5d0
    else if (x <= (-1.75d-299)) then
        tmp = t_0
    else if (x <= 1.85d-265) then
        tmp = y * (-0.5d0)
    else if (x <= 2d+42) 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 - (x / y);
	double tmp;
	if (x <= -7.2e+28) {
		tmp = -1.0;
	} else if (x <= -1.85e-9) {
		tmp = t_0;
	} else if (x <= -4.2e-160) {
		tmp = x * 0.5;
	} else if (x <= -1.75e-299) {
		tmp = t_0;
	} else if (x <= 1.85e-265) {
		tmp = y * -0.5;
	} else if (x <= 2e+42) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if x <= -7.2e+28:
		tmp = -1.0
	elif x <= -1.85e-9:
		tmp = t_0
	elif x <= -4.2e-160:
		tmp = x * 0.5
	elif x <= -1.75e-299:
		tmp = t_0
	elif x <= 1.85e-265:
		tmp = y * -0.5
	elif x <= 2e+42:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (x <= -7.2e+28)
		tmp = -1.0;
	elseif (x <= -1.85e-9)
		tmp = t_0;
	elseif (x <= -4.2e-160)
		tmp = Float64(x * 0.5);
	elseif (x <= -1.75e-299)
		tmp = t_0;
	elseif (x <= 1.85e-265)
		tmp = Float64(y * -0.5);
	elseif (x <= 2e+42)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (x <= -7.2e+28)
		tmp = -1.0;
	elseif (x <= -1.85e-9)
		tmp = t_0;
	elseif (x <= -4.2e-160)
		tmp = x * 0.5;
	elseif (x <= -1.75e-299)
		tmp = t_0;
	elseif (x <= 1.85e-265)
		tmp = y * -0.5;
	elseif (x <= 2e+42)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e+28], -1.0, If[LessEqual[x, -1.85e-9], t$95$0, If[LessEqual[x, -4.2e-160], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, -1.75e-299], t$95$0, If[LessEqual[x, 1.85e-265], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 2e+42], 1.0, -1.0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -7.1999999999999999e28 or 2.00000000000000009e42 < x

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 81.7%

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

   if -7.1999999999999999e28 < x < -1.85e-9 or -4.2000000000000001e-160 < x < -1.74999999999999995e-299

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.7%

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

      3. associate--l- 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around inf 65.4%

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

   7. Taylor expanded in y around 0 65.6%

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

      1. mul-1-neg 65.6%

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

      2. unsub-neg 65.6%

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

   9. Simplified 65.6%

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

   if -1.85e-9 < x < -4.2000000000000001e-160

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 62.5%

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

   5. Taylor expanded in x around 0 60.5%

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

      1. *-commutative 60.5%

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

   7. Simplified 60.5%

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

   if -1.74999999999999995e-299 < x < 1.8499999999999999e-265

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 91.7%

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

      1. mul-1-neg 91.7%

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

      2. distribute-neg-frac 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in y around 0 65.9%

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

      1. *-commutative 65.9%

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

   9. Simplified 65.9%

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

   if 1.8499999999999999e-265 < x < 2.00000000000000009e42

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 67.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-9}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-160}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-299}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-265}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+27}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-160}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4e+27)
   -1.0
   (if (<= x -1.85e-9)
     1.0
     (if (<= x -4.2e-160) (* x 0.5) (if (<= x 4.5e+40) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4e+27) {
		tmp = -1.0;
	} else if (x <= -1.85e-9) {
		tmp = 1.0;
	} else if (x <= -4.2e-160) {
		tmp = x * 0.5;
	} else if (x <= 4.5e+40) {
		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 <= (-4d+27)) then
        tmp = -1.0d0
    else if (x <= (-1.85d-9)) then
        tmp = 1.0d0
    else if (x <= (-4.2d-160)) then
        tmp = x * 0.5d0
    else if (x <= 4.5d+40) 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 <= -4e+27) {
		tmp = -1.0;
	} else if (x <= -1.85e-9) {
		tmp = 1.0;
	} else if (x <= -4.2e-160) {
		tmp = x * 0.5;
	} else if (x <= 4.5e+40) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4e+27:
		tmp = -1.0
	elif x <= -1.85e-9:
		tmp = 1.0
	elif x <= -4.2e-160:
		tmp = x * 0.5
	elif x <= 4.5e+40:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4e+27)
		tmp = -1.0;
	elseif (x <= -1.85e-9)
		tmp = 1.0;
	elseif (x <= -4.2e-160)
		tmp = Float64(x * 0.5);
	elseif (x <= 4.5e+40)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4e+27)
		tmp = -1.0;
	elseif (x <= -1.85e-9)
		tmp = 1.0;
	elseif (x <= -4.2e-160)
		tmp = x * 0.5;
	elseif (x <= 4.5e+40)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4e+27], -1.0, If[LessEqual[x, -1.85e-9], 1.0, If[LessEqual[x, -4.2e-160], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 4.5e+40], 1.0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000001e27 or 4.50000000000000032e40 < x

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 81.0%

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

   if -4.0000000000000001e27 < x < -1.85e-9 or -4.2000000000000001e-160 < x < 4.50000000000000032e40

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 63.4%

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

   if -1.85e-9 < x < -4.2000000000000001e-160

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 62.5%

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

   5. Taylor expanded in x around 0 60.5%

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

      1. *-commutative 60.5%

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

   7. Simplified 60.5%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+27}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-160}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -65000000 \lor \neg \left(y \leq 45000000000000\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -65000000.0) (not (<= y 45000000000000.0)))
   (- 1.0 (/ x y))
   (/ x (- 2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -65000000.0) || !(y <= 45000000000000.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (2.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-65000000.0d0)) .or. (.not. (y <= 45000000000000.0d0))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / (2.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -65000000.0) || !(y <= 45000000000000.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (2.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -65000000.0) or not (y <= 45000000000000.0):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (2.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -65000000.0) || !(y <= 45000000000000.0))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / Float64(2.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -65000000.0) || ~((y <= 45000000000000.0)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (2.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -65000000.0], N[Not[LessEqual[y, 45000000000000.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5e7 or 4.5e13 < y

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.8%

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

      3. associate--l- 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 80.0%

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

   7. Taylor expanded in y around 0 80.1%

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

      1. mul-1-neg 80.1%

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

      2. unsub-neg 80.1%

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

   9. Simplified 80.1%

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

   if -6.5e7 < y < 4.5e13

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 78.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -65000000 \lor \neg \left(y \leq 45000000000000\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 - x}\\ \end{array} \]

Alternative 7: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -27000000000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -27000000000.0)
   (- 1.0 (/ x y))
   (if (<= y 2.6e-7) (/ x (- 2.0 x)) (/ y (+ y -2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -27000000000.0) {
		tmp = 1.0 - (x / y);
	} else if (y <= 2.6e-7) {
		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 <= (-27000000000.0d0)) then
        tmp = 1.0d0 - (x / y)
    else if (y <= 2.6d-7) 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 <= -27000000000.0) {
		tmp = 1.0 - (x / y);
	} else if (y <= 2.6e-7) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -27000000000.0:
		tmp = 1.0 - (x / y)
	elif y <= 2.6e-7:
		tmp = x / (2.0 - x)
	else:
		tmp = y / (y + -2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -27000000000.0)
		tmp = Float64(1.0 - Float64(x / y));
	elseif (y <= 2.6e-7)
		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 <= -27000000000.0)
		tmp = 1.0 - (x / y);
	elseif (y <= 2.6e-7)
		tmp = x / (2.0 - x);
	else
		tmp = y / (y + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -27000000000.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e-7], 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 < -2.7e10

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.8%

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

      3. associate--l- 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 80.0%

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

   7. Taylor expanded in y around 0 80.2%

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

      1. mul-1-neg 80.2%

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

      2. unsub-neg 80.2%

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

   9. Simplified 80.2%

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

   if -2.7e10 < y < 2.59999999999999999e-7

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 79.4%

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

   if 2.59999999999999999e-7 < y

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 79.7%

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

      1. mul-1-neg 79.7%

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

      2. distribute-neg-frac 79.7%

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

   6. Simplified 79.7%

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

      1. expm1-log1p-u 79.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-y}{2 - y}\right)\right)} \]

      2. expm1-udef 79.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-y}{2 - y}\right)} - 1} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{2 - y}\right)} - 1 \]

      4. sqrt-unprod 11.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{2 - y}\right)} - 1 \]

      5. sqr-neg 11.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{y \cdot y}}}{2 - y}\right)} - 1 \]

      6. sqrt-unprod 16.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{2 - y}\right)} - 1 \]

      7. add-sqr-sqrt 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{y}}{2 - y}\right)} - 1 \]

      8. frac-2neg 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-y}{-\left(2 - y\right)}}\right)} - 1 \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{-\left(2 - y\right)}\right)} - 1 \]

      10. sqrt-unprod 30.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{-\left(2 - y\right)}\right)} - 1 \]

      11. sqr-neg 30.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{y \cdot y}}}{-\left(2 - y\right)}\right)} - 1 \]

      12. sqrt-unprod 79.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{-\left(2 - y\right)}\right)} - 1 \]

      13. add-sqr-sqrt 79.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{y}}{-\left(2 - y\right)}\right)} - 1 \]

      14. sub-neg 79.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{-\color{blue}{\left(2 + \left(-y\right)\right)}}\right)} - 1 \]

      15. distribute-neg-in 79.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{\color{blue}{\left(-2\right) + \left(-\left(-y\right)\right)}}\right)} - 1 \]

      16. metadata-eval 79.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{\color{blue}{-2} + \left(-\left(-y\right)\right)}\right)} - 1 \]

      17. remove-double-neg 79.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{-2 + \color{blue}{y}}\right)} - 1 \]

   8. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{-2 + y}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 79.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{-2 + y}\right)\right)} \]

      2. expm1-log1p 79.7%

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

      3. +-commutative 79.7%

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

   10. Simplified 79.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -27000000000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \]

Alternative 8: 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. Final simplification 100.0%

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

Alternative 9: 61.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+39}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.35e+29) -1.0 (if (<= x 3.5e+39) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.35e+29) {
		tmp = -1.0;
	} else if (x <= 3.5e+39) {
		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 <= (-2.35d+29)) then
        tmp = -1.0d0
    else if (x <= 3.5d+39) 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 <= -2.35e+29) {
		tmp = -1.0;
	} else if (x <= 3.5e+39) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.35e+29:
		tmp = -1.0
	elif x <= 3.5e+39:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.35e+29)
		tmp = -1.0;
	elseif (x <= 3.5e+39)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.35e+29)
		tmp = -1.0;
	elseif (x <= 3.5e+39)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.35e+29], -1.0, If[LessEqual[x, 3.5e+39], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3500000000000001e29 or 3.5000000000000002e39 < x

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 81.7%

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

   if -2.3500000000000001e29 < x < 3.5000000000000002e39

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 56.7%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+39}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 36.8% 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. Step-by-step derivation

   1. associate--r+ 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 35.0%

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

5. Final simplification 35.0%

   \[\leadsto -1 \]

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 2023222 
(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))))