Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.3% → 99.4%

Time: 7.3s

Alternatives: 7

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-43} \lor \neg \left(x \leq 5 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{y}{-0.5 \cdot \frac{y}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2e-43) (not (<= x 5e-77)))
   (/ y (+ (* -0.5 (/ y x)) 0.5))
   (/ (* x 2.0) (/ (- x y) y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2e-43) || !(x <= 5e-77)) {
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	} else {
		tmp = (x * 2.0) / ((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 <= (-2d-43)) .or. (.not. (x <= 5d-77))) then
        tmp = y / (((-0.5d0) * (y / x)) + 0.5d0)
    else
        tmp = (x * 2.0d0) / ((x - y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2e-43) || !(x <= 5e-77)) {
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	} else {
		tmp = (x * 2.0) / ((x - y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2e-43) or not (x <= 5e-77):
		tmp = y / ((-0.5 * (y / x)) + 0.5)
	else:
		tmp = (x * 2.0) / ((x - y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2e-43) || !(x <= 5e-77))
		tmp = Float64(y / Float64(Float64(-0.5 * Float64(y / x)) + 0.5));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2e-43) || ~((x <= 5e-77)))
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	else
		tmp = (x * 2.0) / ((x - y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2e-43], N[Not[LessEqual[x, 5e-77]], $MachinePrecision]], N[(y / N[(N[(-0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000015e-43 or 4.99999999999999963e-77 < x

   1. Initial program 79.3%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. clear-num 99.8%

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

      3. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -2.00000000000000015e-43 < x < 4.99999999999999963e-77

   1. Initial program 74.2%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-43} \lor \neg \left(x \leq 5 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{y}{-0.5 \cdot \frac{y}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-130} \lor \neg \left(x \leq 1.06 \cdot 10^{-156}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.5e-130) (not (<= x 1.06e-156)))
   (* y (/ (* x 2.0) (- x y)))
   (* x -2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.5e-130) || !(x <= 1.06e-156)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-5.5d-130)) .or. (.not. (x <= 1.06d-156))) then
        tmp = y * ((x * 2.0d0) / (x - y))
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5.5e-130) || !(x <= 1.06e-156)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.5e-130) or not (x <= 1.06e-156):
		tmp = y * ((x * 2.0) / (x - y))
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.5e-130) || !(x <= 1.06e-156))
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5.5e-130) || ~((x <= 1.06e-156)))
		tmp = y * ((x * 2.0) / (x - y));
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.5e-130], N[Not[LessEqual[x, 1.06e-156]], $MachinePrecision]], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.50000000000000007e-130 or 1.06e-156 < x

   1. Initial program 79.4%

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

      1. associate-*l/ 97.7%

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

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing

   if -5.50000000000000007e-130 < x < 1.06e-156

   1. Initial program 72.0%

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

      1. associate-*l/ 69.2%

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

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 90.7%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-130} \lor \neg \left(x \leq 1.06 \cdot 10^{-156}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-131} \lor \neg \left(x \leq 5.6 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{y}{-0.5 \cdot \frac{y}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -8.4e-131) (not (<= x 5.6e-151)))
   (/ y (+ (* -0.5 (/ y x)) 0.5))
   (* x -2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -8.4e-131) || !(x <= 5.6e-151)) {
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-8.4d-131)) .or. (.not. (x <= 5.6d-151))) then
        tmp = y / (((-0.5d0) * (y / x)) + 0.5d0)
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -8.4e-131) || !(x <= 5.6e-151)) {
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -8.4e-131) or not (x <= 5.6e-151):
		tmp = y / ((-0.5 * (y / x)) + 0.5)
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -8.4e-131) || !(x <= 5.6e-151))
		tmp = Float64(y / Float64(Float64(-0.5 * Float64(y / x)) + 0.5));
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -8.4e-131) || ~((x <= 5.6e-151)))
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -8.4e-131], N[Not[LessEqual[x, 5.6e-151]], $MachinePrecision]], N[(y / N[(N[(-0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.39999999999999988e-131 or 5.6000000000000002e-151 < x

   1. Initial program 79.4%

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

      1. associate-*l/ 97.7%

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

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 97.7%

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

      2. clear-num 97.7%

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

      3. un-div-inv 97.8%

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

   6. Applied egg-rr 97.8%

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

   7. Taylor expanded in x around 0 97.8%

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

      1. +-commutative 97.8%

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

   9. Simplified 97.8%

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

   if -8.39999999999999988e-131 < x < 5.6000000000000002e-151

   1. Initial program 72.0%

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

      1. associate-*l/ 69.2%

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

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 90.7%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-131} \lor \neg \left(x \leq 5.6 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{y}{-0.5 \cdot \frac{y}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-43} \lor \neg \left(x \leq 7.6 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{y}{-0.5 \cdot \frac{y}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{1 - \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.16e-43) (not (<= x 7.6e-76)))
   (/ y (+ (* -0.5 (/ y x)) 0.5))
   (/ (* x -2.0) (- 1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.16e-43) || !(x <= 7.6e-76)) {
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	} else {
		tmp = (x * -2.0) / (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 <= (-1.16d-43)) .or. (.not. (x <= 7.6d-76))) then
        tmp = y / (((-0.5d0) * (y / x)) + 0.5d0)
    else
        tmp = (x * (-2.0d0)) / (1.0d0 - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.16e-43) || !(x <= 7.6e-76)) {
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	} else {
		tmp = (x * -2.0) / (1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.16e-43) or not (x <= 7.6e-76):
		tmp = y / ((-0.5 * (y / x)) + 0.5)
	else:
		tmp = (x * -2.0) / (1.0 - (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.16e-43) || !(x <= 7.6e-76))
		tmp = Float64(y / Float64(Float64(-0.5 * Float64(y / x)) + 0.5));
	else
		tmp = Float64(Float64(x * -2.0) / Float64(1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.16e-43) || ~((x <= 7.6e-76)))
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	else
		tmp = (x * -2.0) / (1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.16e-43], N[Not[LessEqual[x, 7.6e-76]], $MachinePrecision]], N[(y / N[(N[(-0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] / N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1600000000000001e-43 or 7.6000000000000004e-76 < x

   1. Initial program 79.3%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. clear-num 99.8%

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

      3. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -1.1600000000000001e-43 < x < 7.6000000000000004e-76

   1. Initial program 74.2%

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

      1. *-lft-identity 74.2%

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

      2. metadata-eval 74.2%

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

      3. times-frac 74.2%

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

      4. neg-mul-1 74.2%

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

      5. sub-neg 74.2%

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

      6. +-commutative 74.2%

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

      7. distribute-neg-out 74.2%

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

      8. remove-double-neg 74.2%

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

      9. sub-neg 74.2%

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

      10. associate-*r* 74.2%

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

      11. neg-mul-1 74.2%

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

      12. distribute-lft-neg-out 74.2%

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

      13. associate-/l* 99.9%

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

      14. distribute-lft-neg-out 99.9%

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

      15. distribute-rgt-neg-in 99.9%

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

      16. metadata-eval 99.9%

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

      17. div-sub 99.9%

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

      18. *-inverses 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-43} \lor \neg \left(x \leq 7.6 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{y}{-0.5 \cdot \frac{y}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{1 - \frac{x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-131}:\\ \;\;\;\;\frac{y}{\frac{x - y}{x \cdot 2}}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-159}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-0.5 \cdot \frac{y}{x} + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e-131)
   (/ y (/ (- x y) (* x 2.0)))
   (if (<= x 4.5e-159) (* x -2.0) (/ y (+ (* -0.5 (/ y x)) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-131) {
		tmp = y / ((x - y) / (x * 2.0));
	} else if (x <= 4.5e-159) {
		tmp = x * -2.0;
	} else {
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.4d-131)) then
        tmp = y / ((x - y) / (x * 2.0d0))
    else if (x <= 4.5d-159) then
        tmp = x * (-2.0d0)
    else
        tmp = y / (((-0.5d0) * (y / x)) + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-131) {
		tmp = y / ((x - y) / (x * 2.0));
	} else if (x <= 4.5e-159) {
		tmp = x * -2.0;
	} else {
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-131:
		tmp = y / ((x - y) / (x * 2.0))
	elif x <= 4.5e-159:
		tmp = x * -2.0
	else:
		tmp = y / ((-0.5 * (y / x)) + 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-131)
		tmp = Float64(y / Float64(Float64(x - y) / Float64(x * 2.0)));
	elseif (x <= 4.5e-159)
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(y / Float64(Float64(-0.5 * Float64(y / x)) + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e-131)
		tmp = y / ((x - y) / (x * 2.0));
	elseif (x <= 4.5e-159)
		tmp = x * -2.0;
	else
		tmp = y / ((-0.5 * (y / x)) + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-131], N[(y / N[(N[(x - y), $MachinePrecision] / N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-159], N[(x * -2.0), $MachinePrecision], N[(y / N[(N[(-0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3999999999999999e-131

   1. Initial program 79.5%

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

      1. associate-*l/ 97.8%

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

   3. Simplified 97.8%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 97.8%

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

      2. clear-num 97.7%

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

      3. un-div-inv 97.8%

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

   6. Applied egg-rr 97.8%

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

   if -4.3999999999999999e-131 < x < 4.49999999999999989e-159

   1. Initial program 72.0%

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

      1. associate-*l/ 69.2%

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

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 90.7%

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

   if 4.49999999999999989e-159 < x

   1. Initial program 79.3%

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

      1. associate-*l/ 97.7%

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

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 97.7%

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

      2. clear-num 97.7%

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

      3. un-div-inv 97.8%

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

   6. Applied egg-rr 97.8%

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

   7. Taylor expanded in x around 0 97.8%

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

      1. +-commutative 97.8%

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

   9. Simplified 97.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-131}:\\ \;\;\;\;\frac{y}{\frac{x - y}{x \cdot 2}}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-159}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-0.5 \cdot \frac{y}{x} + 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-24} \lor \neg \left(y \leq 1.45 \cdot 10^{-20}\right):\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.1e-24) (not (<= y 1.45e-20))) (* x -2.0) (* y 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.1e-24) || !(y <= 1.45e-20)) {
		tmp = x * -2.0;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3.1d-24)) .or. (.not. (y <= 1.45d-20))) then
        tmp = x * (-2.0d0)
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.1e-24) || !(y <= 1.45e-20)) {
		tmp = x * -2.0;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.1e-24) or not (y <= 1.45e-20):
		tmp = x * -2.0
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.1e-24) || !(y <= 1.45e-20))
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.1e-24) || ~((y <= 1.45e-20)))
		tmp = x * -2.0;
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.1e-24], N[Not[LessEqual[y, 1.45e-20]], $MachinePrecision]], N[(x * -2.0), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1e-24 or 1.45e-20 < y

   1. Initial program 76.6%

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

      1. associate-*l/ 78.0%

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

   3. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 76.5%

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

   if -3.1e-24 < y < 1.45e-20

   1. Initial program 77.6%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 75.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-24} \lor \neg \left(y \leq 1.45 \cdot 10^{-20}\right):\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x -2.0))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x * -2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. associate-*l/ 89.0%

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

3. Simplified 89.0%

   \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 50.9%

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

6. Final simplification 50.9%

   \[\leadsto x \cdot -2 \]
7. Add Preprocessing

Developer target: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (* 2.0 x) (- x y)) y)))
   (if (< x -1.7210442634149447e+81)
     t_0
     (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((2.0 * x) / (x - y)) * y;
	double tmp;
	if (x < -1.7210442634149447e+81) {
		tmp = t_0;
	} else if (x < 83645045635564430.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} 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 = ((2.0d0 * x) / (x - y)) * y
    if (x < (-1.7210442634149447d+81)) then
        tmp = t_0
    else if (x < 83645045635564430.0d0) then
        tmp = (x * 2.0d0) / ((x - y) / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((2.0 * x) / (x - y)) * y;
	double tmp;
	if (x < -1.7210442634149447e+81) {
		tmp = t_0;
	} else if (x < 83645045635564430.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((2.0 * x) / (x - y)) * y
	tmp = 0
	if x < -1.7210442634149447e+81:
		tmp = t_0
	elif x < 83645045635564430.0:
		tmp = (x * 2.0) / ((x - y) / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(2.0 * x) / Float64(x - y)) * y)
	tmp = 0.0
	if (x < -1.7210442634149447e+81)
		tmp = t_0;
	elseif (x < 83645045635564430.0)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((2.0 * x) / (x - y)) * y;
	tmp = 0.0;
	if (x < -1.7210442634149447e+81)
		tmp = t_0;
	elseif (x < 83645045635564430.0)
		tmp = (x * 2.0) / ((x - y) / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(2.0 * x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[Less[x, -1.7210442634149447e+81], t$95$0, If[Less[x, 83645045635564430.0], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024029 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

  (/ (* (* x 2.0) y) (- x y)))