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

Percentage Accurate: 100.0% → 100.0%

Time: 11.7s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 71.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2 - x}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+57}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-81}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+147}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+177}:\\ \;\;\;\;-1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- 2.0 x))))
   (if (<= y -3.1e+57)
     1.0
     (if (<= y 1.6e-150)
       t_0
       (if (<= y 1.9e-81)
         (* (- x y) 0.5)
         (if (<= y 2.35e+58)
           t_0
           (if (<= y 1.9e+147)
             1.0
             (if (<= y 1.2e+177) (+ -1.0 (/ y x)) 1.0))))))))

C

double code(double x, double y) {
	double t_0 = x / (2.0 - x);
	double tmp;
	if (y <= -3.1e+57) {
		tmp = 1.0;
	} else if (y <= 1.6e-150) {
		tmp = t_0;
	} else if (y <= 1.9e-81) {
		tmp = (x - y) * 0.5;
	} else if (y <= 2.35e+58) {
		tmp = t_0;
	} else if (y <= 1.9e+147) {
		tmp = 1.0;
	} else if (y <= 1.2e+177) {
		tmp = -1.0 + (y / 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) :: t_0
    real(8) :: tmp
    t_0 = x / (2.0d0 - x)
    if (y <= (-3.1d+57)) then
        tmp = 1.0d0
    else if (y <= 1.6d-150) then
        tmp = t_0
    else if (y <= 1.9d-81) then
        tmp = (x - y) * 0.5d0
    else if (y <= 2.35d+58) then
        tmp = t_0
    else if (y <= 1.9d+147) then
        tmp = 1.0d0
    else if (y <= 1.2d+177) then
        tmp = (-1.0d0) + (y / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (2.0 - x);
	double tmp;
	if (y <= -3.1e+57) {
		tmp = 1.0;
	} else if (y <= 1.6e-150) {
		tmp = t_0;
	} else if (y <= 1.9e-81) {
		tmp = (x - y) * 0.5;
	} else if (y <= 2.35e+58) {
		tmp = t_0;
	} else if (y <= 1.9e+147) {
		tmp = 1.0;
	} else if (y <= 1.2e+177) {
		tmp = -1.0 + (y / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (2.0 - x)
	tmp = 0
	if y <= -3.1e+57:
		tmp = 1.0
	elif y <= 1.6e-150:
		tmp = t_0
	elif y <= 1.9e-81:
		tmp = (x - y) * 0.5
	elif y <= 2.35e+58:
		tmp = t_0
	elif y <= 1.9e+147:
		tmp = 1.0
	elif y <= 1.2e+177:
		tmp = -1.0 + (y / x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(2.0 - x))
	tmp = 0.0
	if (y <= -3.1e+57)
		tmp = 1.0;
	elseif (y <= 1.6e-150)
		tmp = t_0;
	elseif (y <= 1.9e-81)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 2.35e+58)
		tmp = t_0;
	elseif (y <= 1.9e+147)
		tmp = 1.0;
	elseif (y <= 1.2e+177)
		tmp = Float64(-1.0 + Float64(y / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (2.0 - x);
	tmp = 0.0;
	if (y <= -3.1e+57)
		tmp = 1.0;
	elseif (y <= 1.6e-150)
		tmp = t_0;
	elseif (y <= 1.9e-81)
		tmp = (x - y) * 0.5;
	elseif (y <= 2.35e+58)
		tmp = t_0;
	elseif (y <= 1.9e+147)
		tmp = 1.0;
	elseif (y <= 1.2e+177)
		tmp = -1.0 + (y / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.1e+57], 1.0, If[LessEqual[y, 1.6e-150], t$95$0, If[LessEqual[y, 1.9e-81], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 2.35e+58], t$95$0, If[LessEqual[y, 1.9e+147], 1.0, If[LessEqual[y, 1.2e+177], N[(-1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], 1.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.10000000000000013e57 or 2.34999999999999986e58 < y < 1.89999999999999985e147 or 1.2e177 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 82.2%

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

   if -3.10000000000000013e57 < y < 1.5999999999999999e-150 or 1.8999999999999999e-81 < y < 2.34999999999999986e58

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 80.6%

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

   if 1.5999999999999999e-150 < y < 1.8999999999999999e-81

   1. Initial program 100.0%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 74.3%

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

   6. Taylor expanded in y around 0 74.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(x - y\right) \]

   if 1.89999999999999985e147 < y < 1.2e177

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 83.8%

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

   6. Taylor expanded in x around 0 83.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+57}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-81}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+147}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+177}:\\ \;\;\;\;-1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2 - x}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-81}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{2 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- 2.0 x))))
   (if (<= y -1.15e-16)
     (/ y (+ y -2.0))
     (if (<= y 4.9e-150)
       t_0
       (if (<= y 2.05e-81)
         (* (- x y) 0.5)
         (if (<= y 1.3e+58) t_0 (+ 1.0 (/ (- 2.0 x) y))))))))

C

double code(double x, double y) {
	double t_0 = x / (2.0 - x);
	double tmp;
	if (y <= -1.15e-16) {
		tmp = y / (y + -2.0);
	} else if (y <= 4.9e-150) {
		tmp = t_0;
	} else if (y <= 2.05e-81) {
		tmp = (x - y) * 0.5;
	} else if (y <= 1.3e+58) {
		tmp = t_0;
	} else {
		tmp = 1.0 + ((2.0 - x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (2.0d0 - x)
    if (y <= (-1.15d-16)) then
        tmp = y / (y + (-2.0d0))
    else if (y <= 4.9d-150) then
        tmp = t_0
    else if (y <= 2.05d-81) then
        tmp = (x - y) * 0.5d0
    else if (y <= 1.3d+58) then
        tmp = t_0
    else
        tmp = 1.0d0 + ((2.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (2.0 - x);
	double tmp;
	if (y <= -1.15e-16) {
		tmp = y / (y + -2.0);
	} else if (y <= 4.9e-150) {
		tmp = t_0;
	} else if (y <= 2.05e-81) {
		tmp = (x - y) * 0.5;
	} else if (y <= 1.3e+58) {
		tmp = t_0;
	} else {
		tmp = 1.0 + ((2.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (2.0 - x)
	tmp = 0
	if y <= -1.15e-16:
		tmp = y / (y + -2.0)
	elif y <= 4.9e-150:
		tmp = t_0
	elif y <= 2.05e-81:
		tmp = (x - y) * 0.5
	elif y <= 1.3e+58:
		tmp = t_0
	else:
		tmp = 1.0 + ((2.0 - x) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(2.0 - x))
	tmp = 0.0
	if (y <= -1.15e-16)
		tmp = Float64(y / Float64(y + -2.0));
	elseif (y <= 4.9e-150)
		tmp = t_0;
	elseif (y <= 2.05e-81)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 1.3e+58)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(Float64(2.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (2.0 - x);
	tmp = 0.0;
	if (y <= -1.15e-16)
		tmp = y / (y + -2.0);
	elseif (y <= 4.9e-150)
		tmp = t_0;
	elseif (y <= 2.05e-81)
		tmp = (x - y) * 0.5;
	elseif (y <= 1.3e+58)
		tmp = t_0;
	else
		tmp = 1.0 + ((2.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e-16], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e-150], t$95$0, If[LessEqual[y, 2.05e-81], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 1.3e+58], t$95$0, N[(1.0 + N[(N[(2.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.15e-16

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 78.7%

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

      1. mul-1-neg 78.7%

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

      2. distribute-neg-frac 78.7%

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

   5. Simplified 78.7%

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

      1. expm1-log1p-u 78.7%

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

      2. expm1-udef 76.6%

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

      3. add-sqr-sqrt 76.5%

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

      4. sqrt-unprod 31.7%

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

      5. sqr-neg 31.7%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 19.2%

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

      8. frac-2neg 19.2%

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

      9. add-sqr-sqrt 12.6%

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

      10. sqrt-unprod 12.8%

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

      11. sqr-neg 12.8%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 76.6%

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

      14. sub-neg 76.6%

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

      15. distribute-neg-in 76.6%

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

      16. metadata-eval 76.6%

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

      17. remove-double-neg 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. expm1-def 78.7%

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

      2. expm1-log1p 78.7%

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

      3. +-commutative 78.7%

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

   9. Simplified 78.7%

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

   if -1.15e-16 < y < 4.8999999999999995e-150 or 2.04999999999999992e-81 < y < 1.29999999999999994e58

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 85.4%

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

   if 4.8999999999999995e-150 < y < 2.04999999999999992e-81

   1. Initial program 100.0%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 74.3%

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

   6. Taylor expanded in y around 0 74.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(x - y\right) \]

   if 1.29999999999999994e58 < y

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 76.1%

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

   6. Taylor expanded in y around inf 76.3%

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

      1. +-commutative 76.3%

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

      2. mul-1-neg 76.3%

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

      3. sub-neg 76.3%

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

      4. associate-*r/ 76.3%

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

      5. metadata-eval 76.3%

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

      6. div-sub 76.3%

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

   8. Simplified 76.3%

      \[\leadsto \color{blue}{1 + \frac{2 - x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-81}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{2 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2 - x}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{\frac{y + -2}{y}}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-81}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{2 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- 2.0 x))))
   (if (<= y -3.1e-16)
     (/ 1.0 (/ (+ y -2.0) y))
     (if (<= y 4.9e-150)
       t_0
       (if (<= y 2.1e-81)
         (* (- x y) 0.5)
         (if (<= y 5.8e+58) t_0 (+ 1.0 (/ (- 2.0 x) y))))))))

C

double code(double x, double y) {
	double t_0 = x / (2.0 - x);
	double tmp;
	if (y <= -3.1e-16) {
		tmp = 1.0 / ((y + -2.0) / y);
	} else if (y <= 4.9e-150) {
		tmp = t_0;
	} else if (y <= 2.1e-81) {
		tmp = (x - y) * 0.5;
	} else if (y <= 5.8e+58) {
		tmp = t_0;
	} else {
		tmp = 1.0 + ((2.0 - x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (2.0d0 - x)
    if (y <= (-3.1d-16)) then
        tmp = 1.0d0 / ((y + (-2.0d0)) / y)
    else if (y <= 4.9d-150) then
        tmp = t_0
    else if (y <= 2.1d-81) then
        tmp = (x - y) * 0.5d0
    else if (y <= 5.8d+58) then
        tmp = t_0
    else
        tmp = 1.0d0 + ((2.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (2.0 - x);
	double tmp;
	if (y <= -3.1e-16) {
		tmp = 1.0 / ((y + -2.0) / y);
	} else if (y <= 4.9e-150) {
		tmp = t_0;
	} else if (y <= 2.1e-81) {
		tmp = (x - y) * 0.5;
	} else if (y <= 5.8e+58) {
		tmp = t_0;
	} else {
		tmp = 1.0 + ((2.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (2.0 - x)
	tmp = 0
	if y <= -3.1e-16:
		tmp = 1.0 / ((y + -2.0) / y)
	elif y <= 4.9e-150:
		tmp = t_0
	elif y <= 2.1e-81:
		tmp = (x - y) * 0.5
	elif y <= 5.8e+58:
		tmp = t_0
	else:
		tmp = 1.0 + ((2.0 - x) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(2.0 - x))
	tmp = 0.0
	if (y <= -3.1e-16)
		tmp = Float64(1.0 / Float64(Float64(y + -2.0) / y));
	elseif (y <= 4.9e-150)
		tmp = t_0;
	elseif (y <= 2.1e-81)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 5.8e+58)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(Float64(2.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (2.0 - x);
	tmp = 0.0;
	if (y <= -3.1e-16)
		tmp = 1.0 / ((y + -2.0) / y);
	elseif (y <= 4.9e-150)
		tmp = t_0;
	elseif (y <= 2.1e-81)
		tmp = (x - y) * 0.5;
	elseif (y <= 5.8e+58)
		tmp = t_0;
	else
		tmp = 1.0 + ((2.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.1e-16], N[(1.0 / N[(N[(y + -2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e-150], t$95$0, If[LessEqual[y, 2.1e-81], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 5.8e+58], t$95$0, N[(1.0 + N[(N[(2.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.1000000000000001e-16

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 78.7%

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

      1. mul-1-neg 78.7%

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

      2. distribute-neg-frac 78.7%

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

   5. Simplified 78.7%

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

      1. clear-num 78.7%

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

      2. inv-pow 78.7%

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

      3. frac-2neg 78.7%

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

      4. sub-neg 78.7%

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

      5. distribute-neg-in 78.7%

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

      6. metadata-eval 78.7%

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

      7. remove-double-neg 78.7%

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

      8. remove-double-neg 78.7%

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

   7. Applied egg-rr 78.7%

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

      1. unpow-1 78.7%

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

      2. +-commutative 78.7%

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

   9. Simplified 78.7%

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

   if -3.1000000000000001e-16 < y < 4.8999999999999995e-150 or 2.0999999999999999e-81 < y < 5.80000000000000004e58

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 85.4%

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

   if 4.8999999999999995e-150 < y < 2.0999999999999999e-81

   1. Initial program 100.0%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 74.3%

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

   6. Taylor expanded in y around 0 74.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(x - y\right) \]

   if 5.80000000000000004e58 < y

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 76.1%

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

   6. Taylor expanded in y around inf 76.3%

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

      1. +-commutative 76.3%

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

      2. mul-1-neg 76.3%

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

      3. sub-neg 76.3%

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

      4. associate-*r/ 76.3%

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

      5. metadata-eval 76.3%

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

      6. div-sub 76.3%

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

   8. Simplified 76.3%

      \[\leadsto \color{blue}{1 + \frac{2 - x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{\frac{y + -2}{y}}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-81}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{2 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2 - x}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{\frac{-2}{y} - -1}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{2 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- 2.0 x))))
   (if (<= y -2.8e-16)
     (/ 1.0 (- (/ (- 2.0) y) -1.0))
     (if (<= y 4.9e-150)
       t_0
       (if (<= y 2e-81)
         (* (- x y) 0.5)
         (if (<= y 1.65e+58) t_0 (+ 1.0 (/ (- 2.0 x) y))))))))

C

double code(double x, double y) {
	double t_0 = x / (2.0 - x);
	double tmp;
	if (y <= -2.8e-16) {
		tmp = 1.0 / ((-2.0 / y) - -1.0);
	} else if (y <= 4.9e-150) {
		tmp = t_0;
	} else if (y <= 2e-81) {
		tmp = (x - y) * 0.5;
	} else if (y <= 1.65e+58) {
		tmp = t_0;
	} else {
		tmp = 1.0 + ((2.0 - x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (2.0d0 - x)
    if (y <= (-2.8d-16)) then
        tmp = 1.0d0 / ((-2.0d0 / y) - (-1.0d0))
    else if (y <= 4.9d-150) then
        tmp = t_0
    else if (y <= 2d-81) then
        tmp = (x - y) * 0.5d0
    else if (y <= 1.65d+58) then
        tmp = t_0
    else
        tmp = 1.0d0 + ((2.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (2.0 - x);
	double tmp;
	if (y <= -2.8e-16) {
		tmp = 1.0 / ((-2.0 / y) - -1.0);
	} else if (y <= 4.9e-150) {
		tmp = t_0;
	} else if (y <= 2e-81) {
		tmp = (x - y) * 0.5;
	} else if (y <= 1.65e+58) {
		tmp = t_0;
	} else {
		tmp = 1.0 + ((2.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (2.0 - x)
	tmp = 0
	if y <= -2.8e-16:
		tmp = 1.0 / ((-2.0 / y) - -1.0)
	elif y <= 4.9e-150:
		tmp = t_0
	elif y <= 2e-81:
		tmp = (x - y) * 0.5
	elif y <= 1.65e+58:
		tmp = t_0
	else:
		tmp = 1.0 + ((2.0 - x) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(2.0 - x))
	tmp = 0.0
	if (y <= -2.8e-16)
		tmp = Float64(1.0 / Float64(Float64(Float64(-2.0) / y) - -1.0));
	elseif (y <= 4.9e-150)
		tmp = t_0;
	elseif (y <= 2e-81)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 1.65e+58)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(Float64(2.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (2.0 - x);
	tmp = 0.0;
	if (y <= -2.8e-16)
		tmp = 1.0 / ((-2.0 / y) - -1.0);
	elseif (y <= 4.9e-150)
		tmp = t_0;
	elseif (y <= 2e-81)
		tmp = (x - y) * 0.5;
	elseif (y <= 1.65e+58)
		tmp = t_0;
	else
		tmp = 1.0 + ((2.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.8e-16], N[(1.0 / N[(N[((-2.0) / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e-150], t$95$0, If[LessEqual[y, 2e-81], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 1.65e+58], t$95$0, N[(1.0 + N[(N[(2.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.8000000000000001e-16

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 78.7%

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

      1. mul-1-neg 78.7%

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

      2. div-sub 78.8%

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

      3. metadata-eval 78.8%

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

      4. associate-*r/ 78.8%

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

      5. sub-neg 78.8%

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

      6. associate-*r/ 78.8%

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

      7. metadata-eval 78.8%

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

      8. *-inverses 78.8%

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

      9. metadata-eval 78.8%

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

   9. Simplified 78.8%

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

   if -2.8000000000000001e-16 < y < 4.8999999999999995e-150 or 1.9999999999999999e-81 < y < 1.64999999999999991e58

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 85.4%

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

   if 4.8999999999999995e-150 < y < 1.9999999999999999e-81

   1. Initial program 100.0%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 74.3%

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

   6. Taylor expanded in y around 0 74.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(x - y\right) \]

   if 1.64999999999999991e58 < y

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 76.1%

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

   6. Taylor expanded in y around inf 76.3%

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

      1. +-commutative 76.3%

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

      2. mul-1-neg 76.3%

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

      3. sub-neg 76.3%

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

      4. associate-*r/ 76.3%

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

      5. metadata-eval 76.3%

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

      6. div-sub 76.3%

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

   8. Simplified 76.3%

      \[\leadsto \color{blue}{1 + \frac{2 - x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{\frac{-2}{y} - -1}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{2 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + -2}\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-185}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y -2.0))))
   (if (<= x -4.1e-54)
     (/ x (- 2.0 x))
     (if (<= x 8.6e-197)
       t_0
       (if (<= x 3.9e-185)
         (* (- x y) 0.5)
         (if (<= x 4.2e+29) t_0 (/ (- x y) (- x))))))))

C

double code(double x, double y) {
	double t_0 = y / (y + -2.0);
	double tmp;
	if (x <= -4.1e-54) {
		tmp = x / (2.0 - x);
	} else if (x <= 8.6e-197) {
		tmp = t_0;
	} else if (x <= 3.9e-185) {
		tmp = (x - y) * 0.5;
	} else if (x <= 4.2e+29) {
		tmp = t_0;
	} else {
		tmp = (x - y) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + (-2.0d0))
    if (x <= (-4.1d-54)) then
        tmp = x / (2.0d0 - x)
    else if (x <= 8.6d-197) then
        tmp = t_0
    else if (x <= 3.9d-185) then
        tmp = (x - y) * 0.5d0
    else if (x <= 4.2d+29) then
        tmp = t_0
    else
        tmp = (x - y) / -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (y + -2.0);
	double tmp;
	if (x <= -4.1e-54) {
		tmp = x / (2.0 - x);
	} else if (x <= 8.6e-197) {
		tmp = t_0;
	} else if (x <= 3.9e-185) {
		tmp = (x - y) * 0.5;
	} else if (x <= 4.2e+29) {
		tmp = t_0;
	} else {
		tmp = (x - y) / -x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + -2.0)
	tmp = 0
	if x <= -4.1e-54:
		tmp = x / (2.0 - x)
	elif x <= 8.6e-197:
		tmp = t_0
	elif x <= 3.9e-185:
		tmp = (x - y) * 0.5
	elif x <= 4.2e+29:
		tmp = t_0
	else:
		tmp = (x - y) / -x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + -2.0))
	tmp = 0.0
	if (x <= -4.1e-54)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (x <= 8.6e-197)
		tmp = t_0;
	elseif (x <= 3.9e-185)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 4.2e+29)
		tmp = t_0;
	else
		tmp = Float64(Float64(x - y) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + -2.0);
	tmp = 0.0;
	if (x <= -4.1e-54)
		tmp = x / (2.0 - x);
	elseif (x <= 8.6e-197)
		tmp = t_0;
	elseif (x <= 3.9e-185)
		tmp = (x - y) * 0.5;
	elseif (x <= 4.2e+29)
		tmp = t_0;
	else
		tmp = (x - y) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.1e-54], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.6e-197], t$95$0, If[LessEqual[x, 3.9e-185], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4.2e+29], t$95$0, N[(N[(x - y), $MachinePrecision] / (-x)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.1000000000000001e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.4%

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

   if -4.1000000000000001e-54 < x < 8.6000000000000001e-197 or 3.8999999999999999e-185 < x < 4.2000000000000003e29

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. distribute-neg-frac 81.6%

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

   5. Simplified 81.6%

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

      1. expm1-log1p-u 81.6%

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

      2. expm1-udef 63.1%

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

      3. add-sqr-sqrt 35.6%

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

      4. sqrt-unprod 17.6%

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

      5. sqr-neg 17.6%

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

      6. sqrt-unprod 1.4%

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

      7. add-sqr-sqrt 2.6%

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

      8. frac-2neg 2.6%

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

      9. add-sqr-sqrt 1.0%

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

      10. sqrt-unprod 13.9%

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

      11. sqr-neg 13.9%

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

      12. sqrt-unprod 27.4%

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

      13. add-sqr-sqrt 63.1%

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

      14. sub-neg 63.1%

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

      15. distribute-neg-in 63.1%

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

      16. metadata-eval 63.1%

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

      17. remove-double-neg 63.1%

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

   7. Applied egg-rr 63.1%

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

      1. expm1-def 81.6%

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

      2. expm1-log1p 81.6%

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

      3. +-commutative 81.6%

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

   9. Simplified 81.6%

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

   if 8.6000000000000001e-197 < x < 3.8999999999999999e-185

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(x - y\right) \]

   if 4.2000000000000003e29 < x

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 78.6%

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

      1. expm1-log1p-u 76.5%

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

      2. expm1-udef 76.5%

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

      3. *-commutative 76.5%

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

      4. frac-2neg 76.5%

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

      5. metadata-eval 76.5%

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

      6. un-div-inv 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. expm1-def 76.6%

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

      2. expm1-log1p 78.8%

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

   9. Simplified 78.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-197}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-185}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{y}{x}\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-277}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-118}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ y x))))
   (if (<= x -4.4e+15)
     t_0
     (if (<= x 2.5e-277)
       1.0
       (if (<= x 1.6e-118) (* (- x y) 0.5) (if (<= x 8e+25) 1.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = -1.0 + (y / x);
	double tmp;
	if (x <= -4.4e+15) {
		tmp = t_0;
	} else if (x <= 2.5e-277) {
		tmp = 1.0;
	} else if (x <= 1.6e-118) {
		tmp = (x - y) * 0.5;
	} else if (x <= 8e+25) {
		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) :: tmp
    t_0 = (-1.0d0) + (y / x)
    if (x <= (-4.4d+15)) then
        tmp = t_0
    else if (x <= 2.5d-277) then
        tmp = 1.0d0
    else if (x <= 1.6d-118) then
        tmp = (x - y) * 0.5d0
    else if (x <= 8d+25) 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 = -1.0 + (y / x);
	double tmp;
	if (x <= -4.4e+15) {
		tmp = t_0;
	} else if (x <= 2.5e-277) {
		tmp = 1.0;
	} else if (x <= 1.6e-118) {
		tmp = (x - y) * 0.5;
	} else if (x <= 8e+25) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + (y / x)
	tmp = 0
	if x <= -4.4e+15:
		tmp = t_0
	elif x <= 2.5e-277:
		tmp = 1.0
	elif x <= 1.6e-118:
		tmp = (x - y) * 0.5
	elif x <= 8e+25:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(y / x))
	tmp = 0.0
	if (x <= -4.4e+15)
		tmp = t_0;
	elseif (x <= 2.5e-277)
		tmp = 1.0;
	elseif (x <= 1.6e-118)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 8e+25)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 + (y / x);
	tmp = 0.0;
	if (x <= -4.4e+15)
		tmp = t_0;
	elseif (x <= 2.5e-277)
		tmp = 1.0;
	elseif (x <= 1.6e-118)
		tmp = (x - y) * 0.5;
	elseif (x <= 8e+25)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e+15], t$95$0, If[LessEqual[x, 2.5e-277], 1.0, If[LessEqual[x, 1.6e-118], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 8e+25], 1.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4e15 or 8.00000000000000072e25 < x

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 80.1%

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

   6. Taylor expanded in x around 0 80.2%

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

   if -4.4e15 < x < 2.5e-277 or 1.60000000000000002e-118 < x < 8.00000000000000072e25

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 65.0%

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

   if 2.5e-277 < x < 1.60000000000000002e-118

   1. Initial program 100.0%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in y around 0 67.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+15}:\\ \;\;\;\;-1 + \frac{y}{x}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-277}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-118}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + -2}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-185}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y -2.0))))
   (if (<= x -8e-55)
     (/ x (- 2.0 x))
     (if (<= x 8.6e-197)
       t_0
       (if (<= x 3.9e-185)
         (* (- x y) 0.5)
         (if (<= x 6e+29) t_0 (+ -1.0 (/ y x))))))))

C

double code(double x, double y) {
	double t_0 = y / (y + -2.0);
	double tmp;
	if (x <= -8e-55) {
		tmp = x / (2.0 - x);
	} else if (x <= 8.6e-197) {
		tmp = t_0;
	} else if (x <= 3.9e-185) {
		tmp = (x - y) * 0.5;
	} else if (x <= 6e+29) {
		tmp = t_0;
	} else {
		tmp = -1.0 + (y / x);
	}
	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 = y / (y + (-2.0d0))
    if (x <= (-8d-55)) then
        tmp = x / (2.0d0 - x)
    else if (x <= 8.6d-197) then
        tmp = t_0
    else if (x <= 3.9d-185) then
        tmp = (x - y) * 0.5d0
    else if (x <= 6d+29) then
        tmp = t_0
    else
        tmp = (-1.0d0) + (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (y + -2.0);
	double tmp;
	if (x <= -8e-55) {
		tmp = x / (2.0 - x);
	} else if (x <= 8.6e-197) {
		tmp = t_0;
	} else if (x <= 3.9e-185) {
		tmp = (x - y) * 0.5;
	} else if (x <= 6e+29) {
		tmp = t_0;
	} else {
		tmp = -1.0 + (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + -2.0)
	tmp = 0
	if x <= -8e-55:
		tmp = x / (2.0 - x)
	elif x <= 8.6e-197:
		tmp = t_0
	elif x <= 3.9e-185:
		tmp = (x - y) * 0.5
	elif x <= 6e+29:
		tmp = t_0
	else:
		tmp = -1.0 + (y / x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + -2.0))
	tmp = 0.0
	if (x <= -8e-55)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (x <= 8.6e-197)
		tmp = t_0;
	elseif (x <= 3.9e-185)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 6e+29)
		tmp = t_0;
	else
		tmp = Float64(-1.0 + Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + -2.0);
	tmp = 0.0;
	if (x <= -8e-55)
		tmp = x / (2.0 - x);
	elseif (x <= 8.6e-197)
		tmp = t_0;
	elseif (x <= 3.9e-185)
		tmp = (x - y) * 0.5;
	elseif (x <= 6e+29)
		tmp = t_0;
	else
		tmp = -1.0 + (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-55], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.6e-197], t$95$0, If[LessEqual[x, 3.9e-185], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6e+29], t$95$0, N[(-1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.99999999999999996e-55

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.4%

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

   if -7.99999999999999996e-55 < x < 8.6000000000000001e-197 or 3.8999999999999999e-185 < x < 5.9999999999999998e29

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. distribute-neg-frac 81.6%

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

   5. Simplified 81.6%

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

      1. expm1-log1p-u 81.6%

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

      2. expm1-udef 63.1%

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

      3. add-sqr-sqrt 35.6%

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

      4. sqrt-unprod 17.6%

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

      5. sqr-neg 17.6%

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

      6. sqrt-unprod 1.4%

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

      7. add-sqr-sqrt 2.6%

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

      8. frac-2neg 2.6%

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

      9. add-sqr-sqrt 1.0%

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

      10. sqrt-unprod 13.9%

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

      11. sqr-neg 13.9%

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

      12. sqrt-unprod 27.4%

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

      13. add-sqr-sqrt 63.1%

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

      14. sub-neg 63.1%

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

      15. distribute-neg-in 63.1%

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

      16. metadata-eval 63.1%

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

      17. remove-double-neg 63.1%

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

   7. Applied egg-rr 63.1%

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

      1. expm1-def 81.6%

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

      2. expm1-log1p 81.6%

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

      3. +-commutative 81.6%

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

   9. Simplified 81.6%

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

   if 8.6000000000000001e-197 < x < 3.8999999999999999e-185

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(x - y\right) \]

   if 5.9999999999999998e29 < x

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 78.6%

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

   6. Taylor expanded in x around 0 78.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-197}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-185}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-273}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-173}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+30}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e+15)
   -1.0
   (if (<= x 8e-273)
     1.0
     (if (<= x 4.6e-173) (* y -0.5) (if (<= x 1.85e+30) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.1e+15) {
		tmp = -1.0;
	} else if (x <= 8e-273) {
		tmp = 1.0;
	} else if (x <= 4.6e-173) {
		tmp = y * -0.5;
	} else if (x <= 1.85e+30) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.1d+15)) then
        tmp = -1.0d0
    else if (x <= 8d-273) then
        tmp = 1.0d0
    else if (x <= 4.6d-173) then
        tmp = y * (-0.5d0)
    else if (x <= 1.85d+30) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.1e+15) {
		tmp = -1.0;
	} else if (x <= 8e-273) {
		tmp = 1.0;
	} else if (x <= 4.6e-173) {
		tmp = y * -0.5;
	} else if (x <= 1.85e+30) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.1e+15:
		tmp = -1.0
	elif x <= 8e-273:
		tmp = 1.0
	elif x <= 4.6e-173:
		tmp = y * -0.5
	elif x <= 1.85e+30:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.1e+15)
		tmp = -1.0;
	elseif (x <= 8e-273)
		tmp = 1.0;
	elseif (x <= 4.6e-173)
		tmp = Float64(y * -0.5);
	elseif (x <= 1.85e+30)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.1e+15)
		tmp = -1.0;
	elseif (x <= 8e-273)
		tmp = 1.0;
	elseif (x <= 4.6e-173)
		tmp = y * -0.5;
	elseif (x <= 1.85e+30)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.1e+15], -1.0, If[LessEqual[x, 8e-273], 1.0, If[LessEqual[x, 4.6e-173], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 1.85e+30], 1.0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1e15 or 1.85000000000000008e30 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.7%

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

   if -3.1e15 < x < 8.000000000000001e-273 or 4.59999999999999976e-173 < x < 1.85000000000000008e30

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 64.3%

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

   if 8.000000000000001e-273 < x < 4.59999999999999976e-173

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.0%

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

      1. mul-1-neg 79.0%

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

      2. distribute-neg-frac 79.0%

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

   5. Simplified 79.0%

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

   6. Taylor expanded in y around 0 49.3%

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

      1. *-commutative 49.3%

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

   8. Simplified 49.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-273}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-173}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+30}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-274}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-118}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e+15)
   -1.0
   (if (<= x 7e-274)
     1.0
     (if (<= x 9.5e-118) (* (- x y) 0.5) (if (<= x 2.5e+24) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+15) {
		tmp = -1.0;
	} else if (x <= 7e-274) {
		tmp = 1.0;
	} else if (x <= 9.5e-118) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.5e+24) {
		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 <= (-3.4d+15)) then
        tmp = -1.0d0
    else if (x <= 7d-274) then
        tmp = 1.0d0
    else if (x <= 9.5d-118) then
        tmp = (x - y) * 0.5d0
    else if (x <= 2.5d+24) 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 <= -3.4e+15) {
		tmp = -1.0;
	} else if (x <= 7e-274) {
		tmp = 1.0;
	} else if (x <= 9.5e-118) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.5e+24) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.4e+15:
		tmp = -1.0
	elif x <= 7e-274:
		tmp = 1.0
	elif x <= 9.5e-118:
		tmp = (x - y) * 0.5
	elif x <= 2.5e+24:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.4e+15)
		tmp = -1.0;
	elseif (x <= 7e-274)
		tmp = 1.0;
	elseif (x <= 9.5e-118)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2.5e+24)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.4e+15)
		tmp = -1.0;
	elseif (x <= 7e-274)
		tmp = 1.0;
	elseif (x <= 9.5e-118)
		tmp = (x - y) * 0.5;
	elseif (x <= 2.5e+24)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.4e+15], -1.0, If[LessEqual[x, 7e-274], 1.0, If[LessEqual[x, 9.5e-118], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.5e+24], 1.0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e15 or 2.50000000000000023e24 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.7%

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

   if -3.4e15 < x < 6.99999999999999963e-274 or 9.49999999999999931e-118 < x < 2.50000000000000023e24

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 65.0%

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

   if 6.99999999999999963e-274 < x < 9.49999999999999931e-118

   1. Initial program 100.0%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in y around 0 67.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-274}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-118}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -33:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+30}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1}{2 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -33.0)
   (/ x (- 2.0 x))
   (if (<= x 2.15e+30) (* (- x y) (/ 1.0 (- 2.0 y))) (/ (- x y) (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -33.0) {
		tmp = x / (2.0 - x);
	} else if (x <= 2.15e+30) {
		tmp = (x - y) * (1.0 / (2.0 - y));
	} else {
		tmp = (x - y) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-33.0d0)) then
        tmp = x / (2.0d0 - x)
    else if (x <= 2.15d+30) then
        tmp = (x - y) * (1.0d0 / (2.0d0 - y))
    else
        tmp = (x - y) / -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -33.0) {
		tmp = x / (2.0 - x);
	} else if (x <= 2.15e+30) {
		tmp = (x - y) * (1.0 / (2.0 - y));
	} else {
		tmp = (x - y) / -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -33.0:
		tmp = x / (2.0 - x)
	elif x <= 2.15e+30:
		tmp = (x - y) * (1.0 / (2.0 - y))
	else:
		tmp = (x - y) / -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -33.0)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (x <= 2.15e+30)
		tmp = Float64(Float64(x - y) * Float64(1.0 / Float64(2.0 - y)));
	else
		tmp = Float64(Float64(x - y) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -33.0)
		tmp = x / (2.0 - x);
	elseif (x <= 2.15e+30)
		tmp = (x - y) * (1.0 / (2.0 - y));
	else
		tmp = (x - y) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -33.0], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.15e+30], N[(N[(x - y), $MachinePrecision] * N[(1.0 / N[(2.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -33

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 80.4%

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

   if -33 < x < 2.15e30

   1. Initial program 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.7%

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

   if 2.15e30 < x

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 78.6%

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

      1. expm1-log1p-u 76.5%

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

      2. expm1-udef 76.5%

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

      3. *-commutative 76.5%

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

      4. frac-2neg 76.5%

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

      5. metadata-eval 76.5%

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

      6. un-div-inv 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. expm1-def 76.6%

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

      2. expm1-log1p 78.8%

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

   9. Simplified 78.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -33:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+30}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1}{2 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{+14}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+31}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.4e+14) -1.0 (if (<= x 3.8e+31) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.4e+14) {
		tmp = -1.0;
	} else if (x <= 3.8e+31) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.4d+14)) then
        tmp = -1.0d0
    else if (x <= 3.8d+31) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.4e+14) {
		tmp = -1.0;
	} else if (x <= 3.8e+31) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.4e+14:
		tmp = -1.0
	elif x <= 3.8e+31:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.4e+14)
		tmp = -1.0;
	elseif (x <= 3.8e+31)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.4e+14)
		tmp = -1.0;
	elseif (x <= 3.8e+31)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.4e+14], -1.0, If[LessEqual[x, 3.8e+31], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4e14 or 3.8000000000000001e31 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.7%

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

   if -9.4e14 < x < 3.8000000000000001e31

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 56.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{+14}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+31}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf 39.1%

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

4. Final simplification 39.1%

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

Developer target: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024020 
(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))))