Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.6% → 93.4%

Time: 6.3s

Alternatives: 5

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: 76.6% 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: 93.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+174}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+105}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e+174)
   (* y 2.0)
   (if (<= x 2.6e+105)
     (* (* 2.0 x) (/ y (- x y)))
     (* (fma (/ 2.0 x) y 2.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e+174) {
		tmp = y * 2.0;
	} else if (x <= 2.6e+105) {
		tmp = (2.0 * x) * (y / (x - y));
	} else {
		tmp = fma((2.0 / x), y, 2.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.5e+174)
		tmp = Float64(y * 2.0);
	elseif (x <= 2.6e+105)
		tmp = Float64(Float64(2.0 * x) * Float64(y / Float64(x - y)));
	else
		tmp = Float64(fma(Float64(2.0 / x), y, 2.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e+174], N[(y * 2.0), $MachinePrecision], If[LessEqual[x, 2.6e+105], N[(N[(2.0 * x), $MachinePrecision] * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / x), $MachinePrecision] * y + 2.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5000000000000001e174

   1. Initial program 70.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -3.5000000000000001e174 < x < 2.6000000000000002e105

   1. Initial program 78.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 2.6000000000000002e105 < x

   1. Initial program 74.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+174}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+105}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-2}{y - x} \cdot \left(y \cdot x\right)\\ t_1 := \frac{\left(2 \cdot x\right) \cdot y}{x - y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-299}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;t\_1 \leq 10^{+127}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ -2.0 (- y x)) (* y x))) (t_1 (/ (* (* 2.0 x) y) (- x y))))
   (if (<= t_1 (- INFINITY))
     (* (fma (/ 2.0 x) y 2.0) y)
     (if (<= t_1 -1e-299)
       t_0
       (if (<= t_1 0.0) (* y 2.0) (if (<= t_1 1e+127) t_0 (* -2.0 x)))))))

C

double code(double x, double y) {
	double t_0 = (-2.0 / (y - x)) * (y * x);
	double t_1 = ((2.0 * x) * y) / (x - y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((2.0 / x), y, 2.0) * y;
	} else if (t_1 <= -1e-299) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = y * 2.0;
	} else if (t_1 <= 1e+127) {
		tmp = t_0;
	} else {
		tmp = -2.0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(-2.0 / Float64(y - x)) * Float64(y * x))
	t_1 = Float64(Float64(Float64(2.0 * x) * y) / Float64(x - y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(2.0 / x), y, 2.0) * y);
	elseif (t_1 <= -1e-299)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(y * 2.0);
	elseif (t_1 <= 1e+127)
		tmp = t_0;
	else
		tmp = Float64(-2.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-2.0 / N[(y - x), $MachinePrecision]), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(2.0 / x), $MachinePrecision] * y + 2.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, -1e-299], t$95$0, If[LessEqual[t$95$1, 0.0], N[(y * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+127], t$95$0, N[(-2.0 * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -inf.0

   1. Initial program 4.4%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 79.6

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

   5. Applied rewrites 79.6%

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

   if -inf.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -9.99999999999999992e-300 or -0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 9.99999999999999955e126

   1. Initial program 99.8%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -9.99999999999999992e-300 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -0.0

   1. Initial program 13.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if 9.99999999999999955e126 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 4.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\frac{-2}{y - x} \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq 10^{+127}:\\ \;\;\;\;\frac{-2}{y - x} \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.4e-11)
   (* -2.0 x)
   (if (<= y 8.8e+46) (* (fma (/ 2.0 x) y 2.0) y) (* -2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.4e-11) {
		tmp = -2.0 * x;
	} else if (y <= 8.8e+46) {
		tmp = fma((2.0 / x), y, 2.0) * y;
	} else {
		tmp = -2.0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.4e-11)
		tmp = Float64(-2.0 * x);
	elseif (y <= 8.8e+46)
		tmp = Float64(fma(Float64(2.0 / x), y, 2.0) * y);
	else
		tmp = Float64(-2.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.4e-11], N[(-2.0 * x), $MachinePrecision], If[LessEqual[y, 8.8e+46], N[(N[(N[(2.0 / x), $MachinePrecision] * y + 2.0), $MachinePrecision] * y), $MachinePrecision], N[(-2.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3999999999999999e-11 or 8.8000000000000001e46 < y

   1. Initial program 77.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -3.3999999999999999e-11 < y < 8.8000000000000001e46

   1. Initial program 77.2%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

Alternative 4: 74.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+77}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.4e-11) (* -2.0 x) (if (<= y 2.7e+77) (* y 2.0) (* -2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.4e-11) {
		tmp = -2.0 * x;
	} else if (y <= 2.7e+77) {
		tmp = y * 2.0;
	} else {
		tmp = -2.0 * 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 <= (-3.4d-11)) then
        tmp = (-2.0d0) * x
    else if (y <= 2.7d+77) then
        tmp = y * 2.0d0
    else
        tmp = (-2.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.4e-11) {
		tmp = -2.0 * x;
	} else if (y <= 2.7e+77) {
		tmp = y * 2.0;
	} else {
		tmp = -2.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.4e-11:
		tmp = -2.0 * x
	elif y <= 2.7e+77:
		tmp = y * 2.0
	else:
		tmp = -2.0 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.4e-11)
		tmp = Float64(-2.0 * x);
	elseif (y <= 2.7e+77)
		tmp = Float64(y * 2.0);
	else
		tmp = Float64(-2.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.4e-11)
		tmp = -2.0 * x;
	elseif (y <= 2.7e+77)
		tmp = y * 2.0;
	else
		tmp = -2.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.4e-11], N[(-2.0 * x), $MachinePrecision], If[LessEqual[y, 2.7e+77], N[(y * 2.0), $MachinePrecision], N[(-2.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3999999999999999e-11 or 2.6999999999999998e77 < y

   1. Initial program 77.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

   if -3.3999999999999999e-11 < y < 2.6999999999999998e77

   1. Initial program 77.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

Alternative 5: 50.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 53.9

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

5. Applied rewrites 53.9%

   \[\leadsto \color{blue}{-2 \cdot x} \]
6. 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 2024266 
(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)))