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

Percentage Accurate: 100.0% → 100.0%

Time: 9.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. div-sub 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 74.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + -1\\ t_1 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) -1.0)) (t_1 (/ x (+ x y))))
   (if (<= y -7.5e-21)
     t_0
     (if (<= y 2.1e-32)
       t_1
       (if (<= y 1.35e+104) t_0 (if (<= y 2.7e+122) t_1 (/ (- y) (+ x y))))))))

C

double code(double x, double y) {
	double t_0 = (x / y) + -1.0;
	double t_1 = x / (x + y);
	double tmp;
	if (y <= -7.5e-21) {
		tmp = t_0;
	} else if (y <= 2.1e-32) {
		tmp = t_1;
	} else if (y <= 1.35e+104) {
		tmp = t_0;
	} else if (y <= 2.7e+122) {
		tmp = t_1;
	} else {
		tmp = -y / (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) :: t_1
    real(8) :: tmp
    t_0 = (x / y) + (-1.0d0)
    t_1 = x / (x + y)
    if (y <= (-7.5d-21)) then
        tmp = t_0
    else if (y <= 2.1d-32) then
        tmp = t_1
    else if (y <= 1.35d+104) then
        tmp = t_0
    else if (y <= 2.7d+122) then
        tmp = t_1
    else
        tmp = -y / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) + -1.0;
	double t_1 = x / (x + y);
	double tmp;
	if (y <= -7.5e-21) {
		tmp = t_0;
	} else if (y <= 2.1e-32) {
		tmp = t_1;
	} else if (y <= 1.35e+104) {
		tmp = t_0;
	} else if (y <= 2.7e+122) {
		tmp = t_1;
	} else {
		tmp = -y / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + -1.0
	t_1 = x / (x + y)
	tmp = 0
	if y <= -7.5e-21:
		tmp = t_0
	elif y <= 2.1e-32:
		tmp = t_1
	elif y <= 1.35e+104:
		tmp = t_0
	elif y <= 2.7e+122:
		tmp = t_1
	else:
		tmp = -y / (x + y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + -1.0)
	t_1 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= -7.5e-21)
		tmp = t_0;
	elseif (y <= 2.1e-32)
		tmp = t_1;
	elseif (y <= 1.35e+104)
		tmp = t_0;
	elseif (y <= 2.7e+122)
		tmp = t_1;
	else
		tmp = Float64(Float64(-y) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + -1.0;
	t_1 = x / (x + y);
	tmp = 0.0;
	if (y <= -7.5e-21)
		tmp = t_0;
	elseif (y <= 2.1e-32)
		tmp = t_1;
	elseif (y <= 1.35e+104)
		tmp = t_0;
	elseif (y <= 2.7e+122)
		tmp = t_1;
	else
		tmp = -y / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e-21], t$95$0, If[LessEqual[y, 2.1e-32], t$95$1, If[LessEqual[y, 1.35e+104], t$95$0, If[LessEqual[y, 2.7e+122], t$95$1, N[((-y) / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.50000000000000072e-21 or 2.0999999999999999e-32 < y < 1.34999999999999992e104

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. frac-sub 51.5%

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

      2. associate-/r* 52.9%

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

      3. *-commutative 52.9%

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

   6. Applied egg-rr 52.9%

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

   7. Taylor expanded in x around 0 77.0%

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

      1. neg-mul-1 77.0%

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

   9. Simplified 77.0%

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

   10. Taylor expanded in y around inf 77.5%

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

   if -7.50000000000000072e-21 < y < 2.0999999999999999e-32 or 1.34999999999999992e104 < y < 2.6999999999999998e122

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. frac-sub 55.3%

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

      2. associate-/r* 56.2%

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

      3. *-commutative 56.2%

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

   6. Applied egg-rr 56.2%

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

   7. Taylor expanded in x around inf 80.1%

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

   if 2.6999999999999998e122 < y

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. frac-sub 16.7%

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

      2. associate-/r* 18.9%

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

      3. *-commutative 18.9%

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

   6. Applied egg-rr 18.9%

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

   7. Taylor expanded in x around 0 93.4%

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

      1. neg-mul-1 93.4%

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

   9. Simplified 93.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + -1\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) -1.0)))
   (if (<= y -6.6e-21)
     t_0
     (if (<= y 2.25e-32)
       (+ 1.0 (* -2.0 (/ y x)))
       (if (<= y 1.35e+104)
         t_0
         (if (<= y 2.7e+122) (/ x (+ x y)) (/ (- y) (+ x y))))))))

C

double code(double x, double y) {
	double t_0 = (x / y) + -1.0;
	double tmp;
	if (y <= -6.6e-21) {
		tmp = t_0;
	} else if (y <= 2.25e-32) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else if (y <= 1.35e+104) {
		tmp = t_0;
	} else if (y <= 2.7e+122) {
		tmp = x / (x + y);
	} else {
		tmp = -y / (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 / y) + (-1.0d0)
    if (y <= (-6.6d-21)) then
        tmp = t_0
    else if (y <= 2.25d-32) then
        tmp = 1.0d0 + ((-2.0d0) * (y / x))
    else if (y <= 1.35d+104) then
        tmp = t_0
    else if (y <= 2.7d+122) then
        tmp = x / (x + y)
    else
        tmp = -y / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) + -1.0;
	double tmp;
	if (y <= -6.6e-21) {
		tmp = t_0;
	} else if (y <= 2.25e-32) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else if (y <= 1.35e+104) {
		tmp = t_0;
	} else if (y <= 2.7e+122) {
		tmp = x / (x + y);
	} else {
		tmp = -y / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + -1.0
	tmp = 0
	if y <= -6.6e-21:
		tmp = t_0
	elif y <= 2.25e-32:
		tmp = 1.0 + (-2.0 * (y / x))
	elif y <= 1.35e+104:
		tmp = t_0
	elif y <= 2.7e+122:
		tmp = x / (x + y)
	else:
		tmp = -y / (x + y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + -1.0)
	tmp = 0.0
	if (y <= -6.6e-21)
		tmp = t_0;
	elseif (y <= 2.25e-32)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(y / x)));
	elseif (y <= 1.35e+104)
		tmp = t_0;
	elseif (y <= 2.7e+122)
		tmp = Float64(x / Float64(x + y));
	else
		tmp = Float64(Float64(-y) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + -1.0;
	tmp = 0.0;
	if (y <= -6.6e-21)
		tmp = t_0;
	elseif (y <= 2.25e-32)
		tmp = 1.0 + (-2.0 * (y / x));
	elseif (y <= 1.35e+104)
		tmp = t_0;
	elseif (y <= 2.7e+122)
		tmp = x / (x + y);
	else
		tmp = -y / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, -6.6e-21], t$95$0, If[LessEqual[y, 2.25e-32], N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+104], t$95$0, If[LessEqual[y, 2.7e+122], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], N[((-y) / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.60000000000000018e-21 or 2.25000000000000002e-32 < y < 1.34999999999999992e104

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. frac-sub 51.5%

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

      2. associate-/r* 52.9%

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

      3. *-commutative 52.9%

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

   6. Applied egg-rr 52.9%

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

   7. Taylor expanded in x around 0 77.0%

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

      1. neg-mul-1 77.0%

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

   9. Simplified 77.0%

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

   10. Taylor expanded in y around inf 77.5%

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

   if -6.60000000000000018e-21 < y < 2.25000000000000002e-32

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 80.1%

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

   if 1.34999999999999992e104 < y < 2.6999999999999998e122

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. frac-sub 33.3%

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

      2. associate-/r* 34.9%

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

      3. *-commutative 34.9%

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

   6. Applied egg-rr 34.9%

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

   7. Taylor expanded in x around inf 85.1%

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

   if 2.6999999999999998e122 < y

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. frac-sub 16.7%

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

      2. associate-/r* 18.9%

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

      3. *-commutative 18.9%

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

   6. Applied egg-rr 18.9%

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

   7. Taylor expanded in x around 0 93.4%

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

      1. neg-mul-1 93.4%

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

   9. Simplified 93.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-21} \lor \neg \left(y \leq 2.2 \cdot 10^{-32}\right) \land \left(y \leq 1.35 \cdot 10^{+104} \lor \neg \left(y \leq 2.7 \cdot 10^{+122}\right)\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.5e-21)
         (and (not (<= y 2.2e-32))
              (or (<= y 1.35e+104) (not (<= y 2.7e+122)))))
   (+ (/ x y) -1.0)
   (/ x (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e-21) || (!(y <= 2.2e-32) && ((y <= 1.35e+104) || !(y <= 2.7e+122)))) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = x / (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.5d-21)) .or. (.not. (y <= 2.2d-32)) .and. (y <= 1.35d+104) .or. (.not. (y <= 2.7d+122))) then
        tmp = (x / y) + (-1.0d0)
    else
        tmp = x / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e-21) || (!(y <= 2.2e-32) && ((y <= 1.35e+104) || !(y <= 2.7e+122)))) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = x / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.5e-21) or (not (y <= 2.2e-32) and ((y <= 1.35e+104) or not (y <= 2.7e+122))):
		tmp = (x / y) + -1.0
	else:
		tmp = x / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.5e-21) || (!(y <= 2.2e-32) && ((y <= 1.35e+104) || !(y <= 2.7e+122))))
		tmp = Float64(Float64(x / y) + -1.0);
	else
		tmp = Float64(x / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.5e-21) || (~((y <= 2.2e-32)) && ((y <= 1.35e+104) || ~((y <= 2.7e+122)))))
		tmp = (x / y) + -1.0;
	else
		tmp = x / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.5e-21], And[N[Not[LessEqual[y, 2.2e-32]], $MachinePrecision], Or[LessEqual[y, 1.35e+104], N[Not[LessEqual[y, 2.7e+122]], $MachinePrecision]]]], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.49999999999999987e-21 or 2.2e-32 < y < 1.34999999999999992e104 or 2.6999999999999998e122 < y

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. frac-sub 42.1%

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

      2. associate-/r* 43.7%

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

      3. *-commutative 43.7%

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

   6. Applied egg-rr 43.7%

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

   7. Taylor expanded in x around 0 81.4%

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

      1. neg-mul-1 81.4%

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

   9. Simplified 81.4%

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

   10. Taylor expanded in y around inf 81.8%

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

   if -6.49999999999999987e-21 < y < 2.2e-32 or 1.34999999999999992e104 < y < 2.6999999999999998e122

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. frac-sub 55.3%

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

      2. associate-/r* 56.2%

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

      3. *-commutative 56.2%

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

   6. Applied egg-rr 56.2%

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

   7. Taylor expanded in x around inf 80.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-21} \lor \neg \left(y \leq 2.2 \cdot 10^{-32}\right) \land \left(y \leq 1.35 \cdot 10^{+104} \lor \neg \left(y \leq 2.7 \cdot 10^{+122}\right)\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-21}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+122}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.6e-21)
   -1.0
   (if (<= y 4e-42)
     1.0
     (if (<= y 1.35e+104) -1.0 (if (<= y 2.7e+122) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.6e-21) {
		tmp = -1.0;
	} else if (y <= 4e-42) {
		tmp = 1.0;
	} else if (y <= 1.35e+104) {
		tmp = -1.0;
	} else if (y <= 2.7e+122) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.6d-21)) then
        tmp = -1.0d0
    else if (y <= 4d-42) then
        tmp = 1.0d0
    else if (y <= 1.35d+104) then
        tmp = -1.0d0
    else if (y <= 2.7d+122) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.6e-21) {
		tmp = -1.0;
	} else if (y <= 4e-42) {
		tmp = 1.0;
	} else if (y <= 1.35e+104) {
		tmp = -1.0;
	} else if (y <= 2.7e+122) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.6e-21:
		tmp = -1.0
	elif y <= 4e-42:
		tmp = 1.0
	elif y <= 1.35e+104:
		tmp = -1.0
	elif y <= 2.7e+122:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.6e-21)
		tmp = -1.0;
	elseif (y <= 4e-42)
		tmp = 1.0;
	elseif (y <= 1.35e+104)
		tmp = -1.0;
	elseif (y <= 2.7e+122)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.6e-21)
		tmp = -1.0;
	elseif (y <= 4e-42)
		tmp = 1.0;
	elseif (y <= 1.35e+104)
		tmp = -1.0;
	elseif (y <= 2.7e+122)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.6e-21], -1.0, If[LessEqual[y, 4e-42], 1.0, If[LessEqual[y, 1.35e+104], -1.0, If[LessEqual[y, 2.7e+122], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5999999999999995e-21 or 4.00000000000000015e-42 < y < 1.34999999999999992e104 or 2.6999999999999998e122 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 80.7%

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

   if -7.5999999999999995e-21 < y < 4.00000000000000015e-42 or 1.34999999999999992e104 < y < 2.6999999999999998e122

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 79.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-21}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+122}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{-21}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+122}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.1e-21)
   -1.0
   (if (<= y 2.25e-32)
     (- 1.0 (/ y x))
     (if (<= y 1.35e+104) -1.0 (if (<= y 2.7e+122) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.1e-21) {
		tmp = -1.0;
	} else if (y <= 2.25e-32) {
		tmp = 1.0 - (y / x);
	} else if (y <= 1.35e+104) {
		tmp = -1.0;
	} else if (y <= 2.7e+122) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.1d-21)) then
        tmp = -1.0d0
    else if (y <= 2.25d-32) then
        tmp = 1.0d0 - (y / x)
    else if (y <= 1.35d+104) then
        tmp = -1.0d0
    else if (y <= 2.7d+122) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.1e-21) {
		tmp = -1.0;
	} else if (y <= 2.25e-32) {
		tmp = 1.0 - (y / x);
	} else if (y <= 1.35e+104) {
		tmp = -1.0;
	} else if (y <= 2.7e+122) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.1e-21:
		tmp = -1.0
	elif y <= 2.25e-32:
		tmp = 1.0 - (y / x)
	elif y <= 1.35e+104:
		tmp = -1.0
	elif y <= 2.7e+122:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.1e-21)
		tmp = -1.0;
	elseif (y <= 2.25e-32)
		tmp = Float64(1.0 - Float64(y / x));
	elseif (y <= 1.35e+104)
		tmp = -1.0;
	elseif (y <= 2.7e+122)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.1e-21)
		tmp = -1.0;
	elseif (y <= 2.25e-32)
		tmp = 1.0 - (y / x);
	elseif (y <= 1.35e+104)
		tmp = -1.0;
	elseif (y <= 2.7e+122)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.1e-21], -1.0, If[LessEqual[y, 2.25e-32], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+104], -1.0, If[LessEqual[y, 2.7e+122], 1.0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.10000000000000013e-21 or 2.25000000000000002e-32 < y < 1.34999999999999992e104 or 2.6999999999999998e122 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 81.1%

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

   if -6.10000000000000013e-21 < y < 2.25000000000000002e-32

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. frac-sub 56.4%

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

      2. associate-/r* 57.3%

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

      3. *-commutative 57.3%

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in x around inf 79.8%

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

   8. Taylor expanded in x around inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. unsub-neg 79.2%

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

   10. Simplified 79.2%

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

   if 1.34999999999999992e104 < y < 2.6999999999999998e122

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 83.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{-21}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+122}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-21} \lor \neg \left(y \leq 2 \cdot 10^{-32}\right):\\ \;\;\;\;2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6e-21) (not (<= y 2e-32)))
   (+ (* 2.0 (/ x y)) -1.0)
   (+ 1.0 (* -2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6e-21) || !(y <= 2e-32)) {
		tmp = (2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (-2.0 * (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 ((y <= (-6d-21)) .or. (.not. (y <= 2d-32))) then
        tmp = (2.0d0 * (x / y)) + (-1.0d0)
    else
        tmp = 1.0d0 + ((-2.0d0) * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6e-21) || !(y <= 2e-32)) {
		tmp = (2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (-2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6e-21) or not (y <= 2e-32):
		tmp = (2.0 * (x / y)) + -1.0
	else:
		tmp = 1.0 + (-2.0 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6e-21) || !(y <= 2e-32))
		tmp = Float64(Float64(2.0 * Float64(x / y)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(-2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6e-21) || ~((y <= 2e-32)))
		tmp = (2.0 * (x / y)) + -1.0;
	else
		tmp = 1.0 + (-2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6e-21], N[Not[LessEqual[y, 2e-32]], $MachinePrecision]], N[(N[(2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.99999999999999982e-21 or 2.00000000000000011e-32 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 80.1%

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

   if -5.99999999999999982e-21 < y < 2.00000000000000011e-32

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 80.1%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-21} \lor \neg \left(y \leq 2 \cdot 10^{-32}\right):\\ \;\;\;\;2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-21} \lor \neg \left(y \leq 2.05 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.2e-21) (not (<= y 2.05e-32)))
   (+ (/ x y) -1.0)
   (- 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e-21) || !(y <= 2.05e-32)) {
		tmp = (x / y) + -1.0;
	} 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) :: tmp
    if ((y <= (-6.2d-21)) .or. (.not. (y <= 2.05d-32))) then
        tmp = (x / y) + (-1.0d0)
    else
        tmp = 1.0d0 - (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e-21) || !(y <= 2.05e-32)) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = 1.0 - (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.2e-21) or not (y <= 2.05e-32):
		tmp = (x / y) + -1.0
	else:
		tmp = 1.0 - (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.2e-21) || !(y <= 2.05e-32))
		tmp = Float64(Float64(x / y) + -1.0);
	else
		tmp = Float64(1.0 - Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.2e-21) || ~((y <= 2.05e-32)))
		tmp = (x / y) + -1.0;
	else
		tmp = 1.0 - (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.2e-21], N[Not[LessEqual[y, 2.05e-32]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999997e-21 or 2.04999999999999988e-32 < y

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. frac-sub 41.7%

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

      2. associate-/r* 43.3%

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

      3. *-commutative 43.3%

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

   6. Applied egg-rr 43.3%

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

   7. Taylor expanded in x around 0 78.7%

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

      1. neg-mul-1 78.7%

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

   9. Simplified 78.7%

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

   10. Taylor expanded in y around inf 79.0%

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

   if -6.1999999999999997e-21 < y < 2.04999999999999988e-32

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. frac-sub 56.4%

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

      2. associate-/r* 57.3%

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

      3. *-commutative 57.3%

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in x around inf 79.8%

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

   8. Taylor expanded in x around inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. unsub-neg 79.2%

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

   10. Simplified 79.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-21} \lor \neg \left(y \leq 2.05 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. div-sub 100.0%

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

4. Applied egg-rr 100.0%

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

   1. sub-div 99.9%

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

   2. clear-num 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \frac{1}{\frac{x + y}{x - y}} \]
8. Add Preprocessing

Alternative 10: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 11: 49.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 51.1%

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

4. Final simplification 51.1%

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

Developer target: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024041 
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, D"
  :precision binary64

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

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