Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.7% → 99.5%

Time: 6.9s

Alternatives: 7

Speedup: 0.7×

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: 77.7% 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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;\left({\left(y - x\right)}^{-1} \cdot y\right) \cdot \left(x \cdot -2\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+48}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{x - y} \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e+63)
   (* (* (pow (- y x) -1.0) y) (* x -2.0))
   (if (<= y 1.8e+48)
     (* (* 2.0 y) (/ x (- x y)))
     (* (* (/ 2.0 (- x y)) y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+63) {
		tmp = (pow((y - x), -1.0) * y) * (x * -2.0);
	} else if (y <= 1.8e+48) {
		tmp = (2.0 * y) * (x / (x - y));
	} else {
		tmp = ((2.0 / (x - y)) * 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 <= (-2.5d+63)) then
        tmp = (((y - x) ** (-1.0d0)) * y) * (x * (-2.0d0))
    else if (y <= 1.8d+48) then
        tmp = (2.0d0 * y) * (x / (x - y))
    else
        tmp = ((2.0d0 / (x - y)) * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+63) {
		tmp = (Math.pow((y - x), -1.0) * y) * (x * -2.0);
	} else if (y <= 1.8e+48) {
		tmp = (2.0 * y) * (x / (x - y));
	} else {
		tmp = ((2.0 / (x - y)) * y) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e+63:
		tmp = (math.pow((y - x), -1.0) * y) * (x * -2.0)
	elif y <= 1.8e+48:
		tmp = (2.0 * y) * (x / (x - y))
	else:
		tmp = ((2.0 / (x - y)) * y) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e+63)
		tmp = Float64(Float64((Float64(y - x) ^ -1.0) * y) * Float64(x * -2.0));
	elseif (y <= 1.8e+48)
		tmp = Float64(Float64(2.0 * y) * Float64(x / Float64(x - y)));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(x - y)) * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e+63)
		tmp = (((y - x) ^ -1.0) * y) * (x * -2.0);
	elseif (y <= 1.8e+48)
		tmp = (2.0 * y) * (x / (x - y));
	else
		tmp = ((2.0 / (x - y)) * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e+63], N[(N[(N[Power[N[(y - x), $MachinePrecision], -1.0], $MachinePrecision] * y), $MachinePrecision] * N[(x * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+48], N[(N[(2.0 * y), $MachinePrecision] * N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(x - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.50000000000000005e63

   1. Initial program 81.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. neg-sub0 N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\color{blue}{\left(0 - \left(x - y\right)\right)}}^{-1}\right) \]

      19. lift--.f64 N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\left(0 - \color{blue}{\left(x - y\right)}\right)}^{-1}\right) \]

      20. sub-neg N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\left(0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}^{-1}\right) \]

      21. +-commutative N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}\right)}^{-1}\right) \]

      22. associate--r+ N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x\right)}}^{-1}\right) \]

      23. neg-sub0 N/A

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

      24. remove-double-neg N/A

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

      25. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -2.50000000000000005e63 < y < 1.79999999999999992e48

   1. Initial program 78.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 1.79999999999999992e48 < y

   1. Initial program 85.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. neg-sub0 N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\color{blue}{\left(0 - \left(x - y\right)\right)}}^{-1}\right) \]

      19. lift--.f64 N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\left(0 - \color{blue}{\left(x - y\right)}\right)}^{-1}\right) \]

      20. sub-neg N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\left(0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}^{-1}\right) \]

      21. +-commutative N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}\right)}^{-1}\right) \]

      22. associate--r+ N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x\right)}}^{-1}\right) \]

      23. neg-sub0 N/A

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

      24. remove-double-neg N/A

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

      25. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;\left({\left(y - x\right)}^{-1} \cdot y\right) \cdot \left(x \cdot -2\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+48}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{x - y} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{2}{x - y} \cdot y\right) \cdot x\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+48}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (/ 2.0 (- x y)) y) x)))
   (if (<= y -3.5e+92)
     t_0
     (if (<= y 1.8e+48) (* (* 2.0 y) (/ x (- x y))) t_0))))

C

double code(double x, double y) {
	double t_0 = ((2.0 / (x - y)) * y) * x;
	double tmp;
	if (y <= -3.5e+92) {
		tmp = t_0;
	} else if (y <= 1.8e+48) {
		tmp = (2.0 * y) * (x / (x - 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 - y)) * y) * x
    if (y <= (-3.5d+92)) then
        tmp = t_0
    else if (y <= 1.8d+48) then
        tmp = (2.0d0 * y) * (x / (x - 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 - y)) * y) * x;
	double tmp;
	if (y <= -3.5e+92) {
		tmp = t_0;
	} else if (y <= 1.8e+48) {
		tmp = (2.0 * y) * (x / (x - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((2.0 / (x - y)) * y) * x
	tmp = 0
	if y <= -3.5e+92:
		tmp = t_0
	elif y <= 1.8e+48:
		tmp = (2.0 * y) * (x / (x - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(2.0 / Float64(x - y)) * y) * x)
	tmp = 0.0
	if (y <= -3.5e+92)
		tmp = t_0;
	elseif (y <= 1.8e+48)
		tmp = Float64(Float64(2.0 * y) * Float64(x / Float64(x - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((2.0 / (x - y)) * y) * x;
	tmp = 0.0;
	if (y <= -3.5e+92)
		tmp = t_0;
	elseif (y <= 1.8e+48)
		tmp = (2.0 * y) * (x / (x - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(2.0 / N[(x - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3.5e+92], t$95$0, If[LessEqual[y, 1.8e+48], N[(N[(2.0 * y), $MachinePrecision] * N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.49999999999999986e92 or 1.79999999999999992e48 < y

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. neg-sub0 N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\color{blue}{\left(0 - \left(x - y\right)\right)}}^{-1}\right) \]

      19. lift--.f64 N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\left(0 - \color{blue}{\left(x - y\right)}\right)}^{-1}\right) \]

      20. sub-neg N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\left(0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}^{-1}\right) \]

      21. +-commutative N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}\right)}^{-1}\right) \]

      22. associate--r+ N/A

          \[\leadsto \left(-2 \cdot x\right) \cdot \left(y \cdot {\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x\right)}}^{-1}\right) \]

      23. neg-sub0 N/A

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

      24. remove-double-neg N/A

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

      25. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 99.7%

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

   if -3.49999999999999986e92 < y < 1.79999999999999992e48

   1. Initial program 78.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+92}:\\ \;\;\;\;\left(\frac{2}{x - y} \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+48}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{x - y} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+84}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+106}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+84)
   (* x -2.0)
   (if (<= y 1.1e+106) (* (* 2.0 y) (/ x (- x y))) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+84) {
		tmp = x * -2.0;
	} else if (y <= 1.1e+106) {
		tmp = (2.0 * y) * (x / (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 (y <= (-3.8d+84)) then
        tmp = x * (-2.0d0)
    else if (y <= 1.1d+106) then
        tmp = (2.0d0 * y) * (x / (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 (y <= -3.8e+84) {
		tmp = x * -2.0;
	} else if (y <= 1.1e+106) {
		tmp = (2.0 * y) * (x / (x - y));
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+84:
		tmp = x * -2.0
	elif y <= 1.1e+106:
		tmp = (2.0 * y) * (x / (x - y))
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+84)
		tmp = Float64(x * -2.0);
	elseif (y <= 1.1e+106)
		tmp = Float64(Float64(2.0 * y) * Float64(x / Float64(x - y)));
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.8e+84)
		tmp = x * -2.0;
	elseif (y <= 1.1e+106)
		tmp = (2.0 * y) * (x / (x - y));
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+84], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 1.1e+106], N[(N[(2.0 * y), $MachinePrecision] * N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8000000000000001e84 or 1.09999999999999996e106 < y

   1. Initial program 82.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if -3.8000000000000001e84 < y < 1.09999999999999996e106

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 98.9

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

   4. Applied rewrites 98.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+84}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+106}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right) \cdot -2\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+30)
   (* (fma (/ x y) x x) -2.0)
   (if (<= y 3.5e-22) (* (fma (/ 2.0 x) y 2.0) y) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+30) {
		tmp = fma((x / y), x, x) * -2.0;
	} else if (y <= 3.5e-22) {
		tmp = fma((2.0 / x), y, 2.0) * y;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+30)
		tmp = Float64(fma(Float64(x / y), x, x) * -2.0);
	elseif (y <= 3.5e-22)
		tmp = Float64(fma(Float64(2.0 / x), y, 2.0) * y);
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+30], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[y, 3.5e-22], N[(N[(N[(2.0 / x), $MachinePrecision] * y + 2.0), $MachinePrecision] * y), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4499999999999999e30

   1. Initial program 84.8%

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

   3. Taylor expanded in y around inf

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 88.7

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

   5. Applied rewrites 88.7%

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

   if -1.4499999999999999e30 < y < 3.50000000000000005e-22

   1. Initial program 75.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if 3.50000000000000005e-22 < y

   1. Initial program 86.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right) \cdot -2\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+26}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.1e+26)
   (* x -2.0)
   (if (<= y 3.5e-22) (* (fma (/ 2.0 x) y 2.0) y) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+26) {
		tmp = x * -2.0;
	} else if (y <= 3.5e-22) {
		tmp = fma((2.0 / x), y, 2.0) * y;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.1e+26)
		tmp = Float64(x * -2.0);
	elseif (y <= 3.5e-22)
		tmp = Float64(fma(Float64(2.0 / x), y, 2.0) * y);
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.1e+26], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 3.5e-22], N[(N[(N[(2.0 / x), $MachinePrecision] * y + 2.0), $MachinePrecision] * y), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1e26 or 3.50000000000000005e-22 < y

   1. Initial program 85.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -3.1e26 < y < 3.50000000000000005e-22

   1. Initial program 75.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 78.2

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

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+26}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+26}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+39}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.1e+26) (* x -2.0) (if (<= y 6.5e+39) (* 2.0 y) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+26) {
		tmp = x * -2.0;
	} else if (y <= 6.5e+39) {
		tmp = 2.0 * 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 (y <= (-3.1d+26)) then
        tmp = x * (-2.0d0)
    else if (y <= 6.5d+39) then
        tmp = 2.0d0 * y
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+26) {
		tmp = x * -2.0;
	} else if (y <= 6.5e+39) {
		tmp = 2.0 * y;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.1e+26:
		tmp = x * -2.0
	elif y <= 6.5e+39:
		tmp = 2.0 * y
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.1e+26)
		tmp = Float64(x * -2.0);
	elseif (y <= 6.5e+39)
		tmp = Float64(2.0 * y);
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.1e+26)
		tmp = x * -2.0;
	elseif (y <= 6.5e+39)
		tmp = 2.0 * y;
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.1e+26], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 6.5e+39], N[(2.0 * y), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1e26 or 6.5000000000000001e39 < y

   1. Initial program 85.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 87.5

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

   5. Applied rewrites 87.5%

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

   if -3.1e26 < y < 6.5000000000000001e39

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+26}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+39}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.5% accurate, 4.2× 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 80.3%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 54.1

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

5. Applied rewrites 54.1%

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

6. Final simplification 54.1%

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

Developer Target 1: 99.4% accurate, 0.6× 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 2024276 
(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)))