Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.7% → 94.2%

Time: 6.6s

Alternatives: 6

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: 76.7% 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: 94.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y}{x}, y, y\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+178}:\\ \;\;\;\;t\_0 \cdot 2\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+163}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, t\_0, y\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ y x) y y)))
   (if (<= x -2.4e+178)
     (* t_0 2.0)
     (if (<= x 1.7e+163)
       (* (* 2.0 x) (/ y (- x y)))
       (* (fma (/ y x) t_0 y) 2.0)))))

C

double code(double x, double y) {
	double t_0 = fma((y / x), y, y);
	double tmp;
	if (x <= -2.4e+178) {
		tmp = t_0 * 2.0;
	} else if (x <= 1.7e+163) {
		tmp = (2.0 * x) * (y / (x - y));
	} else {
		tmp = fma((y / x), t_0, y) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y / x), y, y)
	tmp = 0.0
	if (x <= -2.4e+178)
		tmp = Float64(t_0 * 2.0);
	elseif (x <= 1.7e+163)
		tmp = Float64(Float64(2.0 * x) * Float64(y / Float64(x - y)));
	else
		tmp = Float64(fma(Float64(y / x), t_0, y) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] * y + y), $MachinePrecision]}, If[LessEqual[x, -2.4e+178], N[(t$95$0 * 2.0), $MachinePrecision], If[LessEqual[x, 1.7e+163], N[(N[(2.0 * x), $MachinePrecision] * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * t$95$0 + y), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4e178

   1. Initial program 50.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y + \left(2 \cdot \frac{{y}^{2}}{x} + 2 \cdot \frac{{y}^{3}}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot y + \color{blue}{2 \cdot \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right) \cdot 2} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   if -2.4e178 < x < 1.7000000000000001e163

   1. Initial program 80.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 96.0

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

      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 96.0

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

   4. Applied rewrites 96.0%

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

   if 1.7000000000000001e163 < x

   1. Initial program 62.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y + \left(2 \cdot \frac{{y}^{2}}{x} + 2 \cdot \frac{{y}^{3}}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot y + \color{blue}{2 \cdot \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right) \cdot 2} \]

   5. Applied rewrites 87.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, y, y\right) \cdot 2\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+163}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \mathsf{fma}\left(\frac{y}{x}, y, y\right), y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y}{x}, y, y\right) \cdot 2\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+163}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (/ y x) y y) 2.0)))
   (if (<= x -2.4e+178)
     t_0
     (if (<= x 1.7e+163) (* (* 2.0 x) (/ y (- x y))) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((y / x), y, y) * 2.0;
	double tmp;
	if (x <= -2.4e+178) {
		tmp = t_0;
	} else if (x <= 1.7e+163) {
		tmp = (2.0 * x) * (y / (x - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(y / x), y, y) * 2.0)
	tmp = 0.0
	if (x <= -2.4e+178)
		tmp = t_0;
	elseif (x <= 1.7e+163)
		tmp = Float64(Float64(2.0 * x) * Float64(y / Float64(x - y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y / x), $MachinePrecision] * y + y), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[x, -2.4e+178], t$95$0, If[LessEqual[x, 1.7e+163], N[(N[(2.0 * x), $MachinePrecision] * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e178 or 1.7000000000000001e163 < x

   1. Initial program 56.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y + \left(2 \cdot \frac{{y}^{2}}{x} + 2 \cdot \frac{{y}^{3}}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot y + \color{blue}{2 \cdot \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right) \cdot 2} \]

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 93.8%

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

   if -2.4e178 < x < 1.7000000000000001e163

   1. Initial program 80.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 96.0

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

      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 96.0

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

   4. Applied rewrites 96.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, y, y\right) \cdot 2\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+163}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, y, y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y}{x}, y, y\right) \cdot 2\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, -2, -2\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (/ y x) y y) 2.0)))
   (if (<= x -4.5e-64)
     t_0
     (if (<= x 1.06e-28) (* (fma (/ x y) -2.0 -2.0) x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((y / x), y, y) * 2.0;
	double tmp;
	if (x <= -4.5e-64) {
		tmp = t_0;
	} else if (x <= 1.06e-28) {
		tmp = fma((x / y), -2.0, -2.0) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(y / x), y, y) * 2.0)
	tmp = 0.0
	if (x <= -4.5e-64)
		tmp = t_0;
	elseif (x <= 1.06e-28)
		tmp = Float64(fma(Float64(x / y), -2.0, -2.0) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y / x), $MachinePrecision] * y + y), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[x, -4.5e-64], t$95$0, If[LessEqual[x, 1.06e-28], N[(N[(N[(x / y), $MachinePrecision] * -2.0 + -2.0), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5000000000000001e-64 or 1.06e-28 < x

   1. Initial program 74.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y + \left(2 \cdot \frac{{y}^{2}}{x} + 2 \cdot \frac{{y}^{3}}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot y + \color{blue}{2 \cdot \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right) \cdot 2} \]

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.4%

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

   if -4.5000000000000001e-64 < x < 1.06e-28

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

Alternative 4: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y}{x}, y, y\right) \cdot 2\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-29}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (/ y x) y y) 2.0)))
   (if (<= x -4.5e-64) t_0 (if (<= x 5.2e-29) (* -2.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((y / x), y, y) * 2.0;
	double tmp;
	if (x <= -4.5e-64) {
		tmp = t_0;
	} else if (x <= 5.2e-29) {
		tmp = -2.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(y / x), y, y) * 2.0)
	tmp = 0.0
	if (x <= -4.5e-64)
		tmp = t_0;
	elseif (x <= 5.2e-29)
		tmp = Float64(-2.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y / x), $MachinePrecision] * y + y), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[x, -4.5e-64], t$95$0, If[LessEqual[x, 5.2e-29], N[(-2.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5000000000000001e-64 or 5.2000000000000004e-29 < x

   1. Initial program 74.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y + \left(2 \cdot \frac{{y}^{2}}{x} + 2 \cdot \frac{{y}^{3}}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot y + \color{blue}{2 \cdot \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y + \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)\right) \cdot 2} \]

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.4%

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

   if -4.5000000000000001e-64 < x < 5.2000000000000004e-29

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

Alternative 5: 73.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-66}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-29}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e-66) (* y 2.0) (if (<= x 5.2e-29) (* -2.0 x) (* y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-66) {
		tmp = y * 2.0;
	} else if (x <= 5.2e-29) {
		tmp = -2.0 * x;
	} 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 (x <= (-2.6d-66)) then
        tmp = y * 2.0d0
    else if (x <= 5.2d-29) then
        tmp = (-2.0d0) * x
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-66) {
		tmp = y * 2.0;
	} else if (x <= 5.2e-29) {
		tmp = -2.0 * x;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.6e-66:
		tmp = y * 2.0
	elif x <= 5.2e-29:
		tmp = -2.0 * x
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.6e-66)
		tmp = Float64(y * 2.0);
	elseif (x <= 5.2e-29)
		tmp = Float64(-2.0 * x);
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6e-66)
		tmp = y * 2.0;
	elseif (x <= 5.2e-29)
		tmp = -2.0 * x;
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.6e-66], N[(y * 2.0), $MachinePrecision], If[LessEqual[x, 5.2e-29], N[(-2.0 * x), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5999999999999999e-66 or 5.2000000000000004e-29 < x

   1. Initial program 74.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -2.5999999999999999e-66 < x < 5.2000000000000004e-29

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-66}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-29}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.5% 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 74.7%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 46.9

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

5. Applied rewrites 46.9%

   \[\leadsto \color{blue}{-2 \cdot x} \]
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 2024332 
(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)))