Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.4% → 99.9%

Time: 7.1s

Alternatives: 5

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: 77.4% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{y - x} \cdot \frac{y}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+29)
   (/ (* x 2.0) (+ (/ x y) -1.0))
   (if (<= y 3e+32) (* (/ x (- y x)) (/ y -0.5)) (* x (* 2.0 (/ y (- x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+29) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else if (y <= 3e+32) {
		tmp = (x / (y - x)) * (y / -0.5);
	} else {
		tmp = x * (2.0 * (y / (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 <= (-1.2d+29)) then
        tmp = (x * 2.0d0) / ((x / y) + (-1.0d0))
    else if (y <= 3d+32) then
        tmp = (x / (y - x)) * (y / (-0.5d0))
    else
        tmp = x * (2.0d0 * (y / (x - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+29) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else if (y <= 3e+32) {
		tmp = (x / (y - x)) * (y / -0.5);
	} else {
		tmp = x * (2.0 * (y / (x - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e+29:
		tmp = (x * 2.0) / ((x / y) + -1.0)
	elif y <= 3e+32:
		tmp = (x / (y - x)) * (y / -0.5)
	else:
		tmp = x * (2.0 * (y / (x - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+29)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0));
	elseif (y <= 3e+32)
		tmp = Float64(Float64(x / Float64(y - x)) * Float64(y / -0.5));
	else
		tmp = Float64(x * Float64(2.0 * Float64(y / Float64(x - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e+29)
		tmp = (x * 2.0) / ((x / y) + -1.0);
	elseif (y <= 3e+32)
		tmp = (x / (y - x)) * (y / -0.5);
	else
		tmp = x * (2.0 * (y / (x - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e+29], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+32], N[(N[(x / N[(y - x), $MachinePrecision]), $MachinePrecision] * N[(y / -0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e29

   1. Initial program 76.2%

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

   3. Taylor expanded in y around inf 76.2%

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

      1. associate-/l* 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/r* 99.9%

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

      5. *-inverses 99.9%

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

      6. associate-*l/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

   7. Simplified 100.0%

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

   if -1.2e29 < y < 3e32

   1. Initial program 74.3%

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

      1. sub-neg 74.3%

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

      2. +-commutative 74.3%

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

      3. neg-sub0 74.3%

         \[\leadsto \frac{\left(x \cdot 2\right) \cdot y}{\color{blue}{\left(0 - y\right)} + x} \]

      4. associate-+l- 74.3%

         \[\leadsto \frac{\left(x \cdot 2\right) \cdot y}{\color{blue}{0 - \left(y - x\right)}} \]

      5. sub0-neg 74.3%

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

      6. distribute-frac-neg2 74.3%

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

      7. distribute-frac-neg 74.3%

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

      8. *-commutative 74.3%

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

      9. associate-*r* 74.3%

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

      10. distribute-rgt-neg-in 74.3%

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

      11. associate-/l* 74.1%

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

      12. *-commutative 74.1%

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

      13. metadata-eval 74.1%

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

   3. Simplified 74.1%

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

      1. clear-num 74.1%

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

      2. un-div-inv 74.3%

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

      3. div-inv 74.3%

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

      4. metadata-eval 74.3%

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

   6. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(y - x\right) \cdot -0.5}} \]
   7. Step-by-step derivation

      1. times-frac 99.9%

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

   8. Simplified 99.9%

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

   if 3e32 < y

   1. Initial program 76.2%

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

      1. associate-/l* 99.9%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-61} \lor \neg \left(y \leq 2.8 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - x} \cdot \frac{y}{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e-61) (not (<= y 2.8e+32)))
   (* x (* 2.0 (/ y (- x y))))
   (* (/ x (- y x)) (/ y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5e-61) || !(y <= 2.8e+32)) {
		tmp = x * (2.0 * (y / (x - y)));
	} else {
		tmp = (x / (y - x)) * (y / -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 ((y <= (-5d-61)) .or. (.not. (y <= 2.8d+32))) then
        tmp = x * (2.0d0 * (y / (x - y)))
    else
        tmp = (x / (y - x)) * (y / (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5e-61) || !(y <= 2.8e+32)) {
		tmp = x * (2.0 * (y / (x - y)));
	} else {
		tmp = (x / (y - x)) * (y / -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5e-61) or not (y <= 2.8e+32):
		tmp = x * (2.0 * (y / (x - y)))
	else:
		tmp = (x / (y - x)) * (y / -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5e-61) || !(y <= 2.8e+32))
		tmp = Float64(x * Float64(2.0 * Float64(y / Float64(x - y))));
	else
		tmp = Float64(Float64(x / Float64(y - x)) * Float64(y / -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5e-61) || ~((y <= 2.8e+32)))
		tmp = x * (2.0 * (y / (x - y)));
	else
		tmp = (x / (y - x)) * (y / -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5e-61], N[Not[LessEqual[y, 2.8e+32]], $MachinePrecision]], N[(x * N[(2.0 * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - x), $MachinePrecision]), $MachinePrecision] * N[(y / -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999999e-61 or 2.8e32 < y

   1. Initial program 78.7%

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

      1. associate-/l* 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   if -4.9999999999999999e-61 < y < 2.8e32

   1. Initial program 71.5%

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

      1. sub-neg 71.5%

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

      2. +-commutative 71.5%

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

      3. neg-sub0 71.5%

         \[\leadsto \frac{\left(x \cdot 2\right) \cdot y}{\color{blue}{\left(0 - y\right)} + x} \]

      4. associate-+l- 71.5%

         \[\leadsto \frac{\left(x \cdot 2\right) \cdot y}{\color{blue}{0 - \left(y - x\right)}} \]

      5. sub0-neg 71.5%

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

      6. distribute-frac-neg2 71.5%

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

      7. distribute-frac-neg 71.5%

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

      8. *-commutative 71.5%

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

      9. associate-*r* 71.5%

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

      10. distribute-rgt-neg-in 71.5%

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

      11. associate-/l* 71.3%

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

      12. *-commutative 71.3%

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

      13. metadata-eval 71.3%

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

   3. Simplified 71.3%

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

      1. clear-num 71.3%

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

      2. un-div-inv 71.5%

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

      3. div-inv 71.5%

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

      4. metadata-eval 71.5%

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

   6. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(y - x\right) \cdot -0.5}} \]
   7. Step-by-step derivation

      1. times-frac 99.9%

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

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x}{y - x} \cdot \frac{y}{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-61} \lor \neg \left(y \leq 2.8 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - x} \cdot \frac{y}{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-160} \lor \neg \left(y \leq 2.4 \cdot 10^{-181}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.1e-160) (not (<= y 2.4e-181)))
   (* x (* 2.0 (/ y (- x y))))
   (* y 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.1e-160) || !(y <= 2.4e-181)) {
		tmp = x * (2.0 * (y / (x - y)));
	} 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 <= (-1.1d-160)) .or. (.not. (y <= 2.4d-181))) then
        tmp = x * (2.0d0 * (y / (x - y)))
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.1e-160) || !(y <= 2.4e-181)) {
		tmp = x * (2.0 * (y / (x - y)));
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.1e-160) or not (y <= 2.4e-181):
		tmp = x * (2.0 * (y / (x - y)))
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.1e-160) || !(y <= 2.4e-181))
		tmp = Float64(x * Float64(2.0 * Float64(y / Float64(x - y))));
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.1e-160) || ~((y <= 2.4e-181)))
		tmp = x * (2.0 * (y / (x - y)));
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.1e-160], N[Not[LessEqual[y, 2.4e-181]], $MachinePrecision]], N[(x * N[(2.0 * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e-160 or 2.4000000000000001e-181 < y

   1. Initial program 78.3%

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

      1. associate-/l* 97.8%

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

      2. associate-*l* 97.8%

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

   3. Simplified 97.8%

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

   if -1.1e-160 < y < 2.4000000000000001e-181

   1. Initial program 64.4%

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

      1. associate-/l* 66.0%

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

      2. associate-*l* 66.0%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in x around inf 93.3%

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

      1. *-commutative 93.3%

         \[\leadsto \color{blue}{y \cdot 2} \]

   7. Simplified 93.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-160} \lor \neg \left(y \leq 2.4 \cdot 10^{-181}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-12} \lor \neg \left(y \leq 4.3 \cdot 10^{-103}\right):\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.75e-12) (not (<= y 4.3e-103))) (* x -2.0) (* y 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.75e-12) || !(y <= 4.3e-103)) {
		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 <= (-1.75d-12)) .or. (.not. (y <= 4.3d-103))) 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 <= -1.75e-12) || !(y <= 4.3e-103)) {
		tmp = x * -2.0;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.75e-12) or not (y <= 4.3e-103):
		tmp = x * -2.0
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.75e-12) || !(y <= 4.3e-103))
		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 <= -1.75e-12) || ~((y <= 4.3e-103)))
		tmp = x * -2.0;
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.75e-12], N[Not[LessEqual[y, 4.3e-103]], $MachinePrecision]], N[(x * -2.0), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.75e-12 or 4.30000000000000023e-103 < y

   1. Initial program 79.0%

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

      1. associate-/l* 99.0%

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

      2. associate-*l* 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around inf 78.7%

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

   if -1.75e-12 < y < 4.30000000000000023e-103

   1. Initial program 69.6%

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

      1. associate-/l* 78.6%

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

      2. associate-*l* 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in x around inf 79.3%

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

      1. *-commutative 79.3%

         \[\leadsto \color{blue}{y \cdot 2} \]

   7. Simplified 79.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-12} \lor \neg \left(y \leq 4.3 \cdot 10^{-103}\right):\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.2% 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 75.2%

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

   1. associate-/l* 90.7%

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

   2. associate-*l* 90.7%

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

3. Simplified 90.7%

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

5. Taylor expanded in y around inf 56.4%

   \[\leadsto x \cdot \color{blue}{-2} \]
6. Add Preprocessing

Developer Target 1: 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 2024186 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1721044263414944700000000000000000000000000000000000000000000000000000000000000000) (* (/ (* 2 x) (- x y)) y) (if (< x 83645045635564430) (/ (* x 2) (/ (- x y) y)) (* (/ (* 2 x) (- x y)) y))))

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