Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 99.9%

Time: 7.4s

Alternatives: 10

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: 99.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(\frac{x + y}{x - y}\right)}^{3}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (cbrt (pow (/ (+ x y) (- x y)) 3.0)))

C

double code(double x, double y) {
	return cbrt(pow(((x + y) / (x - y)), 3.0));
}

Java

public static double code(double x, double y) {
	return Math.cbrt(Math.pow(((x + y) / (x - y)), 3.0));
}

Julia

function code(x, y)
	return cbrt((Float64(Float64(x + y) / Float64(x - y)) ^ 3.0))
end

Wolfram

code[x_, y_] := N[Power[N[Power[N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. add-cbrt-cube 99.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{x + y}{x - y} \cdot \frac{x + y}{x - y}\right) \cdot \frac{x + y}{x - y}}} \]

   2. pow3 99.9%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x + y}{x - y}\right)}^{3}}} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{x + y}{x - y}\right)}^{3}}} \]
5. Add Preprocessing

Alternative 2: 75.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+26} \lor \neg \left(x \leq 6.5 \cdot 10^{+37}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.2e+26) (not (<= x 6.5e+37)))
   (+ 1.0 (* 2.0 (/ y x)))
   (+ (* -2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.2e+26) || !(x <= 6.5e+37)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (-2.0 * (x / y)) + -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 <= (-4.2d+26)) .or. (.not. (x <= 6.5d+37))) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4.2e+26) || !(x <= 6.5e+37)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (-2.0 * (x / y)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.2e+26) or not (x <= 6.5e+37):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = (-2.0 * (x / y)) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.2e+26) || !(x <= 6.5e+37))
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4.2e+26) || ~((x <= 6.5e+37)))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = (-2.0 * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.2e+26], N[Not[LessEqual[x, 6.5e+37]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2000000000000002e26 or 6.4999999999999998e37 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 80.7%

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

   if -4.2000000000000002e26 < x < 6.4999999999999998e37

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+26} \lor \neg \left(x \leq 6.5 \cdot 10^{+37}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+25} \lor \neg \left(x \leq 2.8 \cdot 10^{+37}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{-y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7.5e+25) (not (<= x 2.8e+37)))
   (+ 1.0 (* 2.0 (/ y x)))
   (/ (+ x y) (- y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7.5e+25) || !(x <= 2.8e+37)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (x + y) / -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-7.5d+25)) .or. (.not. (x <= 2.8d+37))) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = (x + y) / -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -7.5e+25) || !(x <= 2.8e+37)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (x + y) / -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -7.5e+25) or not (x <= 2.8e+37):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = (x + y) / -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7.5e+25) || !(x <= 2.8e+37))
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = Float64(Float64(x + y) / Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -7.5e+25) || ~((x <= 2.8e+37)))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = (x + y) / -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7.5e+25], N[Not[LessEqual[x, 2.8e+37]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999993e25 or 2.7999999999999998e37 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 80.7%

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

   if -7.49999999999999993e25 < x < 2.7999999999999998e37

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 72.7%

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

      1. neg-mul-1 72.7%

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

   5. Simplified 72.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+25} \lor \neg \left(x \leq 2.8 \cdot 10^{+37}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+26} \lor \neg \left(x \leq 5.2 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{-y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.4e+26) (not (<= x 5.2e+37)))
   (/ x (- x y))
   (/ (+ x y) (- y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.4e+26) || !(x <= 5.2e+37)) {
		tmp = x / (x - y);
	} else {
		tmp = (x + y) / -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.4d+26)) .or. (.not. (x <= 5.2d+37))) then
        tmp = x / (x - y)
    else
        tmp = (x + y) / -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.4e+26) || !(x <= 5.2e+37)) {
		tmp = x / (x - y);
	} else {
		tmp = (x + y) / -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.4e+26) or not (x <= 5.2e+37):
		tmp = x / (x - y)
	else:
		tmp = (x + y) / -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.4e+26) || !(x <= 5.2e+37))
		tmp = Float64(x / Float64(x - y));
	else
		tmp = Float64(Float64(x + y) / Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.4e+26) || ~((x <= 5.2e+37)))
		tmp = x / (x - y);
	else
		tmp = (x + y) / -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.4e+26], N[Not[LessEqual[x, 5.2e+37]], $MachinePrecision]], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e26 or 5.1999999999999998e37 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 80.5%

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

   if -1.4e26 < x < 5.1999999999999998e37

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 72.7%

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

      1. neg-mul-1 72.7%

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

   5. Simplified 72.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+26} \lor \neg \left(x \leq 5.2 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+25} \lor \neg \left(x \leq 10^{+40}\right):\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.6e+25) (not (<= x 1e+40))) (/ x (- x y)) (- -1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.6e+25) || !(x <= 1e+40)) {
		tmp = x / (x - y);
	} else {
		tmp = -1.0 - (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 ((x <= (-3.6d+25)) .or. (.not. (x <= 1d+40))) then
        tmp = x / (x - y)
    else
        tmp = (-1.0d0) - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.6e+25) || !(x <= 1e+40)) {
		tmp = x / (x - y);
	} else {
		tmp = -1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.6e+25) or not (x <= 1e+40):
		tmp = x / (x - y)
	else:
		tmp = -1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.6e+25) || !(x <= 1e+40))
		tmp = Float64(x / Float64(x - y));
	else
		tmp = Float64(-1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.6e+25) || ~((x <= 1e+40)))
		tmp = x / (x - y);
	else
		tmp = -1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.6e+25], N[Not[LessEqual[x, 1e+40]], $MachinePrecision]], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000015e25 or 1.00000000000000003e40 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 80.5%

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

   if -3.60000000000000015e25 < x < 1.00000000000000003e40

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 72.5%

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

   4. Taylor expanded in y around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. neg-sub0 72.7%

         \[\leadsto \color{blue}{\left(0 - \frac{x}{y}\right)} - 1 \]

      3. associate--r+ 72.7%

         \[\leadsto \color{blue}{0 - \left(\frac{x}{y} + 1\right)} \]

      4. +-commutative 72.7%

         \[\leadsto 0 - \color{blue}{\left(1 + \frac{x}{y}\right)} \]

      5. associate--r+ 72.7%

         \[\leadsto \color{blue}{\left(0 - 1\right) - \frac{x}{y}} \]

      6. metadata-eval 72.7%

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

   6. Simplified 72.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+25} \lor \neg \left(x \leq 10^{+40}\right):\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+26} \lor \neg \left(x \leq 1.4 \cdot 10^{+37}\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.9e+26) (not (<= x 1.4e+37)))
   (+ 1.0 (/ y x))
   (- -1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+26) || !(x <= 1.4e+37)) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0 - (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 ((x <= (-2.9d+26)) .or. (.not. (x <= 1.4d+37))) then
        tmp = 1.0d0 + (y / x)
    else
        tmp = (-1.0d0) - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+26) || !(x <= 1.4e+37)) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.9e+26) or not (x <= 1.4e+37):
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.9e+26) || !(x <= 1.4e+37))
		tmp = Float64(1.0 + Float64(y / x));
	else
		tmp = Float64(-1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.9e+26) || ~((x <= 1.4e+37)))
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.9e+26], N[Not[LessEqual[x, 1.4e+37]], $MachinePrecision]], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e26 or 1.3999999999999999e37 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 80.5%

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

   4. Taylor expanded in x around inf 80.3%

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

   if -2.9e26 < x < 1.3999999999999999e37

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 72.5%

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

   4. Taylor expanded in y around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. neg-sub0 72.7%

         \[\leadsto \color{blue}{\left(0 - \frac{x}{y}\right)} - 1 \]

      3. associate--r+ 72.7%

         \[\leadsto \color{blue}{0 - \left(\frac{x}{y} + 1\right)} \]

      4. +-commutative 72.7%

         \[\leadsto 0 - \color{blue}{\left(1 + \frac{x}{y}\right)} \]

      5. associate--r+ 72.7%

         \[\leadsto \color{blue}{\left(0 - 1\right) - \frac{x}{y}} \]

      6. metadata-eval 72.7%

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

   6. Simplified 72.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+26} \lor \neg \left(x \leq 1.4 \cdot 10^{+37}\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+25} \lor \neg \left(x \leq 1.8 \cdot 10^{-44}\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.8e+25) (not (<= x 1.8e-44))) (+ 1.0 (/ y x)) -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.8e+25) || !(x <= 1.8e-44)) {
		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) :: tmp
    if ((x <= (-3.8d+25)) .or. (.not. (x <= 1.8d-44))) 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 tmp;
	if ((x <= -3.8e+25) || !(x <= 1.8e-44)) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.8e+25) or not (x <= 1.8e-44):
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.8e+25) || !(x <= 1.8e-44))
		tmp = Float64(1.0 + Float64(y / x));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.8e+25) || ~((x <= 1.8e-44)))
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.8e+25], N[Not[LessEqual[x, 1.8e-44]], $MachinePrecision]], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e25 or 1.7999999999999999e-44 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 77.9%

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

   4. Taylor expanded in x around inf 77.8%

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

   if -3.8e25 < x < 1.7999999999999999e-44

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+25} \lor \neg \left(x \leq 1.8 \cdot 10^{-44}\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-43}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e+25) 1.0 (if (<= x 2.3e-43) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e+25) {
		tmp = 1.0;
	} else if (x <= 2.3e-43) {
		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 <= (-3.6d+25)) then
        tmp = 1.0d0
    else if (x <= 2.3d-43) 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 <= -3.6e+25) {
		tmp = 1.0;
	} else if (x <= 2.3e-43) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e+25:
		tmp = 1.0
	elif x <= 2.3e-43:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e+25)
		tmp = 1.0;
	elseif (x <= 2.3e-43)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.6e+25)
		tmp = 1.0;
	elseif (x <= 2.3e-43)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e+25], 1.0, If[LessEqual[x, 2.3e-43], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000015e25 or 2.2999999999999999e-43 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 77.3%

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

   if -3.60000000000000015e25 < x < 2.2999999999999999e-43

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.9%

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

Alternative 9: 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. Add Preprocessing

Alternative 10: 49.3% 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 46.4%

   \[\leadsto \color{blue}{-1} \]
4. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (- (/ x (+ x y)) (/ y (+ x y)))))

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