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

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

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: 62.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot 0.5\\ t_1 := \frac{y}{x} + -1\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-279}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.82 \cdot 10^{+68}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) 0.5)) (t_1 (+ (/ y x) -1.0)))
   (if (<= x -5.8e+24)
     t_1
     (if (<= x -1.05e-32)
       1.0
       (if (<= x 8e-279)
         t_0
         (if (<= x 9e-110)
           1.0
           (if (<= x 3.5e-55) t_0 (if (<= x 1.82e+68) 1.0 t_1))))))))

C

double code(double x, double y) {
	double t_0 = (x - y) * 0.5;
	double t_1 = (y / x) + -1.0;
	double tmp;
	if (x <= -5.8e+24) {
		tmp = t_1;
	} else if (x <= -1.05e-32) {
		tmp = 1.0;
	} else if (x <= 8e-279) {
		tmp = t_0;
	} else if (x <= 9e-110) {
		tmp = 1.0;
	} else if (x <= 3.5e-55) {
		tmp = t_0;
	} else if (x <= 1.82e+68) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	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 = (x - y) * 0.5d0
    t_1 = (y / x) + (-1.0d0)
    if (x <= (-5.8d+24)) then
        tmp = t_1
    else if (x <= (-1.05d-32)) then
        tmp = 1.0d0
    else if (x <= 8d-279) then
        tmp = t_0
    else if (x <= 9d-110) then
        tmp = 1.0d0
    else if (x <= 3.5d-55) then
        tmp = t_0
    else if (x <= 1.82d+68) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) * 0.5;
	double t_1 = (y / x) + -1.0;
	double tmp;
	if (x <= -5.8e+24) {
		tmp = t_1;
	} else if (x <= -1.05e-32) {
		tmp = 1.0;
	} else if (x <= 8e-279) {
		tmp = t_0;
	} else if (x <= 9e-110) {
		tmp = 1.0;
	} else if (x <= 3.5e-55) {
		tmp = t_0;
	} else if (x <= 1.82e+68) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) * 0.5
	t_1 = (y / x) + -1.0
	tmp = 0
	if x <= -5.8e+24:
		tmp = t_1
	elif x <= -1.05e-32:
		tmp = 1.0
	elif x <= 8e-279:
		tmp = t_0
	elif x <= 9e-110:
		tmp = 1.0
	elif x <= 3.5e-55:
		tmp = t_0
	elif x <= 1.82e+68:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * 0.5)
	t_1 = Float64(Float64(y / x) + -1.0)
	tmp = 0.0
	if (x <= -5.8e+24)
		tmp = t_1;
	elseif (x <= -1.05e-32)
		tmp = 1.0;
	elseif (x <= 8e-279)
		tmp = t_0;
	elseif (x <= 9e-110)
		tmp = 1.0;
	elseif (x <= 3.5e-55)
		tmp = t_0;
	elseif (x <= 1.82e+68)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) * 0.5;
	t_1 = (y / x) + -1.0;
	tmp = 0.0;
	if (x <= -5.8e+24)
		tmp = t_1;
	elseif (x <= -1.05e-32)
		tmp = 1.0;
	elseif (x <= 8e-279)
		tmp = t_0;
	elseif (x <= 9e-110)
		tmp = 1.0;
	elseif (x <= 3.5e-55)
		tmp = t_0;
	elseif (x <= 1.82e+68)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -5.8e+24], t$95$1, If[LessEqual[x, -1.05e-32], 1.0, If[LessEqual[x, 8e-279], t$95$0, If[LessEqual[x, 9e-110], 1.0, If[LessEqual[x, 3.5e-55], t$95$0, If[LessEqual[x, 1.82e+68], 1.0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.79999999999999958e24 or 1.81999999999999991e68 < 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 81.6%

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

   7. Taylor expanded in x around 0 81.9%

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

   if -5.79999999999999958e24 < x < -1.05e-32 or 8.00000000000000044e-279 < x < 9.0000000000000002e-110 or 3.50000000000000025e-55 < x < 1.81999999999999991e68

   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 62.6%

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

   if -1.05e-32 < x < 8.00000000000000044e-279 or 9.0000000000000002e-110 < x < 3.50000000000000025e-55

   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 99.8%

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

      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 x around 0 99.8%

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

   7. Taylor expanded in y around 0 64.2%

      \[\leadsto \color{blue}{0.5} \cdot \left(x - y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{x} + -1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-279}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-55}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.82 \cdot 10^{+68}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \]

Alternative 3: 62.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot 0.5\\ \mathbf{if}\;x \leq -5.9 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.26 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-280}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) 0.5)))
   (if (<= x -5.9e+23)
     -1.0
     (if (<= x -1.26e-33)
       1.0
       (if (<= x 3.1e-280)
         t_0
         (if (<= x 5.2e-109)
           1.0
           (if (<= x 2.5e-55) t_0 (if (<= x 3.15e+63) 1.0 -1.0))))))))

C

double code(double x, double y) {
	double t_0 = (x - y) * 0.5;
	double tmp;
	if (x <= -5.9e+23) {
		tmp = -1.0;
	} else if (x <= -1.26e-33) {
		tmp = 1.0;
	} else if (x <= 3.1e-280) {
		tmp = t_0;
	} else if (x <= 5.2e-109) {
		tmp = 1.0;
	} else if (x <= 2.5e-55) {
		tmp = t_0;
	} else if (x <= 3.15e+63) {
		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 - y) * 0.5d0
    if (x <= (-5.9d+23)) then
        tmp = -1.0d0
    else if (x <= (-1.26d-33)) then
        tmp = 1.0d0
    else if (x <= 3.1d-280) then
        tmp = t_0
    else if (x <= 5.2d-109) then
        tmp = 1.0d0
    else if (x <= 2.5d-55) then
        tmp = t_0
    else if (x <= 3.15d+63) 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 - y) * 0.5;
	double tmp;
	if (x <= -5.9e+23) {
		tmp = -1.0;
	} else if (x <= -1.26e-33) {
		tmp = 1.0;
	} else if (x <= 3.1e-280) {
		tmp = t_0;
	} else if (x <= 5.2e-109) {
		tmp = 1.0;
	} else if (x <= 2.5e-55) {
		tmp = t_0;
	} else if (x <= 3.15e+63) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) * 0.5
	tmp = 0
	if x <= -5.9e+23:
		tmp = -1.0
	elif x <= -1.26e-33:
		tmp = 1.0
	elif x <= 3.1e-280:
		tmp = t_0
	elif x <= 5.2e-109:
		tmp = 1.0
	elif x <= 2.5e-55:
		tmp = t_0
	elif x <= 3.15e+63:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * 0.5)
	tmp = 0.0
	if (x <= -5.9e+23)
		tmp = -1.0;
	elseif (x <= -1.26e-33)
		tmp = 1.0;
	elseif (x <= 3.1e-280)
		tmp = t_0;
	elseif (x <= 5.2e-109)
		tmp = 1.0;
	elseif (x <= 2.5e-55)
		tmp = t_0;
	elseif (x <= 3.15e+63)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) * 0.5;
	tmp = 0.0;
	if (x <= -5.9e+23)
		tmp = -1.0;
	elseif (x <= -1.26e-33)
		tmp = 1.0;
	elseif (x <= 3.1e-280)
		tmp = t_0;
	elseif (x <= 5.2e-109)
		tmp = 1.0;
	elseif (x <= 2.5e-55)
		tmp = t_0;
	elseif (x <= 3.15e+63)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -5.9e+23], -1.0, If[LessEqual[x, -1.26e-33], 1.0, If[LessEqual[x, 3.1e-280], t$95$0, If[LessEqual[x, 5.2e-109], 1.0, If[LessEqual[x, 2.5e-55], t$95$0, If[LessEqual[x, 3.15e+63], 1.0, -1.0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.89999999999999987e23 or 3.1499999999999999e63 < 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 81.5%

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

   if -5.89999999999999987e23 < x < -1.26000000000000005e-33 or 3.10000000000000021e-280 < x < 5.1999999999999997e-109 or 2.5000000000000001e-55 < x < 3.1499999999999999e63

   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 62.6%

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

   if -1.26000000000000005e-33 < x < 3.10000000000000021e-280 or 5.1999999999999997e-109 < x < 2.5000000000000001e-55

   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 99.8%

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

      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 x around 0 99.8%

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

   7. Taylor expanded in y around 0 64.2%

      \[\leadsto \color{blue}{0.5} \cdot \left(x - y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.26 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-280}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -78:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -78.0)
   (/ x (- 2.0 x))
   (if (<= x 7.5e+68) (/ (- x y) (- 2.0 y)) (+ (/ y x) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -78.0) {
		tmp = x / (2.0 - x);
	} else if (x <= 7.5e+68) {
		tmp = (x - y) / (2.0 - y);
	} else {
		tmp = (y / x) + -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 <= (-78.0d0)) then
        tmp = x / (2.0d0 - x)
    else if (x <= 7.5d+68) then
        tmp = (x - y) / (2.0d0 - y)
    else
        tmp = (y / x) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -78.0) {
		tmp = x / (2.0 - x);
	} else if (x <= 7.5e+68) {
		tmp = (x - y) / (2.0 - y);
	} else {
		tmp = (y / x) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -78.0:
		tmp = x / (2.0 - x)
	elif x <= 7.5e+68:
		tmp = (x - y) / (2.0 - y)
	else:
		tmp = (y / x) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -78.0)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (x <= 7.5e+68)
		tmp = Float64(Float64(x - y) / Float64(2.0 - y));
	else
		tmp = Float64(Float64(y / x) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -78.0)
		tmp = x / (2.0 - x);
	elseif (x <= 7.5e+68)
		tmp = (x - y) / (2.0 - y);
	else
		tmp = (y / x) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -78.0], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+68], N[(N[(x - y), $MachinePrecision] / N[(2.0 - y), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -78

   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 75.4%

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

   if -78 < x < 7.49999999999999959e68

   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/ 99.8%

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

      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 x around 0 95.6%

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

   7. Taylor expanded in x around 0 95.7%

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

      1. +-commutative 95.7%

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

      2. mul-1-neg 95.7%

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

      3. sub-neg 95.7%

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

      4. div-sub 95.7%

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

   9. Simplified 95.7%

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

   if 7.49999999999999959e68 < 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/ 99.6%

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

      3. associate--l- 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around inf 86.5%

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

   7. Taylor expanded in x around 0 86.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -78:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \]

Alternative 5: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-130}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-280}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+64}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+23)
   -1.0
   (if (<= x -1.55e-130)
     1.0
     (if (<= x 8.6e-280) (* y -0.5) (if (<= x 1.28e+64) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e+23) {
		tmp = -1.0;
	} else if (x <= -1.55e-130) {
		tmp = 1.0;
	} else if (x <= 8.6e-280) {
		tmp = y * -0.5;
	} else if (x <= 1.28e+64) {
		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 <= (-1d+23)) then
        tmp = -1.0d0
    else if (x <= (-1.55d-130)) then
        tmp = 1.0d0
    else if (x <= 8.6d-280) then
        tmp = y * (-0.5d0)
    else if (x <= 1.28d+64) 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 <= -1e+23) {
		tmp = -1.0;
	} else if (x <= -1.55e-130) {
		tmp = 1.0;
	} else if (x <= 8.6e-280) {
		tmp = y * -0.5;
	} else if (x <= 1.28e+64) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e+23:
		tmp = -1.0
	elif x <= -1.55e-130:
		tmp = 1.0
	elif x <= 8.6e-280:
		tmp = y * -0.5
	elif x <= 1.28e+64:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e+23)
		tmp = -1.0;
	elseif (x <= -1.55e-130)
		tmp = 1.0;
	elseif (x <= 8.6e-280)
		tmp = Float64(y * -0.5);
	elseif (x <= 1.28e+64)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e+23)
		tmp = -1.0;
	elseif (x <= -1.55e-130)
		tmp = 1.0;
	elseif (x <= 8.6e-280)
		tmp = y * -0.5;
	elseif (x <= 1.28e+64)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e+23], -1.0, If[LessEqual[x, -1.55e-130], 1.0, If[LessEqual[x, 8.6e-280], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 1.28e+64], 1.0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999992e22 or 1.28000000000000004e64 < 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 81.5%

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

   if -9.9999999999999992e22 < x < -1.55000000000000005e-130 or 8.5999999999999997e-280 < x < 1.28000000000000004e64

   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 56.0%

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

   if -1.55000000000000005e-130 < x < 8.5999999999999997e-280

   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 x around 0 86.4%

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

      1. mul-1-neg 86.4%

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

      2. distribute-neg-frac 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in y around 0 51.8%

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

      1. *-commutative 51.8%

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

   9. Simplified 51.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-130}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-280}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+64}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e+62) 1.0 (if (<= y 4.4e+88) (/ x (- 2.0 x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e+62) {
		tmp = 1.0;
	} else if (y <= 4.4e+88) {
		tmp = x / (2.0 - x);
	} 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 <= (-6d+62)) then
        tmp = 1.0d0
    else if (y <= 4.4d+88) then
        tmp = x / (2.0d0 - x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6e+62) {
		tmp = 1.0;
	} else if (y <= 4.4e+88) {
		tmp = x / (2.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6e+62:
		tmp = 1.0
	elif y <= 4.4e+88:
		tmp = x / (2.0 - x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e+62)
		tmp = 1.0;
	elseif (y <= 4.4e+88)
		tmp = Float64(x / Float64(2.0 - x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6e+62)
		tmp = 1.0;
	elseif (y <= 4.4e+88)
		tmp = x / (2.0 - x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e+62], 1.0, If[LessEqual[y, 4.4e+88], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -6e62 or 4.40000000000000017e88 < 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 y around inf 81.1%

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

   if -6e62 < y < 4.40000000000000017e88

   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 73.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -270:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{+68}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -270.0)
   (/ x (- 2.0 x))
   (if (<= x 2.85e+68) (/ y (+ y -2.0)) (+ (/ y x) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -270.0) {
		tmp = x / (2.0 - x);
	} else if (x <= 2.85e+68) {
		tmp = y / (y + -2.0);
	} else {
		tmp = (y / x) + -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 <= (-270.0d0)) then
        tmp = x / (2.0d0 - x)
    else if (x <= 2.85d+68) then
        tmp = y / (y + (-2.0d0))
    else
        tmp = (y / x) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -270.0) {
		tmp = x / (2.0 - x);
	} else if (x <= 2.85e+68) {
		tmp = y / (y + -2.0);
	} else {
		tmp = (y / x) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -270.0:
		tmp = x / (2.0 - x)
	elif x <= 2.85e+68:
		tmp = y / (y + -2.0)
	else:
		tmp = (y / x) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -270.0)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (x <= 2.85e+68)
		tmp = Float64(y / Float64(y + -2.0));
	else
		tmp = Float64(Float64(y / x) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -270.0)
		tmp = x / (2.0 - x);
	elseif (x <= 2.85e+68)
		tmp = y / (y + -2.0);
	else
		tmp = (y / x) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -270.0], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.85e+68], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -270

   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 75.4%

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

   if -270 < x < 2.8499999999999998e68

   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 x around 0 75.8%

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

      1. mul-1-neg 75.8%

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

      2. distribute-neg-frac 75.8%

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

   6. Simplified 75.8%

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

      1. frac-2neg 75.8%

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

      2. div-inv 75.7%

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

      3. remove-double-neg 75.7%

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

      4. sub-neg 75.7%

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

      5. distribute-neg-in 75.7%

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

      6. metadata-eval 75.7%

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

      7. remove-double-neg 75.7%

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

   8. Applied egg-rr 75.7%

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

      1. associate-*r/ 75.8%

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

      2. *-rgt-identity 75.8%

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

      3. +-commutative 75.8%

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

   10. Simplified 75.8%

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

   if 2.8499999999999998e68 < 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/ 99.6%

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

      3. associate--l- 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around inf 86.5%

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

   7. Taylor expanded in x around 0 86.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -270:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{+68}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \]

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: 62.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+21}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4e+21) -1.0 (if (<= x 2.3e+63) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4e+21) {
		tmp = -1.0;
	} else if (x <= 2.3e+63) {
		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+21)) then
        tmp = -1.0d0
    else if (x <= 2.3d+63) 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+21) {
		tmp = -1.0;
	} else if (x <= 2.3e+63) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4e+21:
		tmp = -1.0
	elif x <= 2.3e+63:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4e+21)
		tmp = -1.0;
	elseif (x <= 2.3e+63)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4e+21)
		tmp = -1.0;
	elseif (x <= 2.3e+63)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4e+21], -1.0, If[LessEqual[x, 2.3e+63], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -4e21 or 2.29999999999999993e63 < 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 81.5%

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

   if -4e21 < x < 2.29999999999999993e63

   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 50.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+21}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 38.3% 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 41.5%

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

5. Final simplification 41.5%

   \[\leadsto -1 \]

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 2023252 
(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))))