Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.5% → 97.4%

Time: 6.0s

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: 77.5% 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: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 2}{\frac{x}{y} - 1}\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-182}:\\ \;\;\;\;\frac{x}{x - y} \cdot \left(2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x 2.0) (- (/ x y) 1.0))))
   (if (<= y -5.1e-86)
     t_0
     (if (<= y 2.75e-182) (* (/ x (- x y)) (* 2.0 y)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * 2.0) / ((x / y) - 1.0);
	double tmp;
	if (y <= -5.1e-86) {
		tmp = t_0;
	} else if (y <= 2.75e-182) {
		tmp = (x / (x - y)) * (2.0 * 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 = (x * 2.0d0) / ((x / y) - 1.0d0)
    if (y <= (-5.1d-86)) then
        tmp = t_0
    else if (y <= 2.75d-182) then
        tmp = (x / (x - y)) * (2.0d0 * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * 2.0) / ((x / y) - 1.0);
	double tmp;
	if (y <= -5.1e-86) {
		tmp = t_0;
	} else if (y <= 2.75e-182) {
		tmp = (x / (x - y)) * (2.0 * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * 2.0) / ((x / y) - 1.0)
	tmp = 0
	if y <= -5.1e-86:
		tmp = t_0
	elif y <= 2.75e-182:
		tmp = (x / (x - y)) * (2.0 * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * 2.0) / Float64(Float64(x / y) - 1.0))
	tmp = 0.0
	if (y <= -5.1e-86)
		tmp = t_0;
	elseif (y <= 2.75e-182)
		tmp = Float64(Float64(x / Float64(x - y)) * Float64(2.0 * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * 2.0) / ((x / y) - 1.0);
	tmp = 0.0;
	if (y <= -5.1e-86)
		tmp = t_0;
	elseif (y <= 2.75e-182)
		tmp = (x / (x - y)) * (2.0 * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.1e-86], t$95$0, If[LessEqual[y, 2.75e-182], N[(N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(2.0 * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.10000000000000006e-86 or 2.74999999999999996e-182 < y

   1. Initial program 75.1%

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

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. *-inverses N/A

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

      6. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if -5.10000000000000006e-86 < y < 2.74999999999999996e-182

   1. Initial program 73.2%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 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 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} - 1}\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-182}:\\ \;\;\;\;\frac{x}{x - y} \cdot \left(2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+244}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{x - y} \cdot \left(2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.15e+244)
   (* -2.0 (fma (/ x y) x x))
   (if (<= y 3.8e+167) (* (/ x (- x y)) (* 2.0 y)) (* -2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.15e+244) {
		tmp = -2.0 * fma((x / y), x, x);
	} else if (y <= 3.8e+167) {
		tmp = (x / (x - y)) * (2.0 * y);
	} else {
		tmp = -2.0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.15e+244)
		tmp = Float64(-2.0 * fma(Float64(x / y), x, x));
	elseif (y <= 3.8e+167)
		tmp = Float64(Float64(x / Float64(x - y)) * Float64(2.0 * y));
	else
		tmp = Float64(-2.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.15e+244], N[(-2.0 * N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+167], N[(N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(2.0 * y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1499999999999999e244

   1. Initial program 42.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-2 \cdot x + -2 \cdot \frac{{x}^{2}}{y}} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

   if -2.1499999999999999e244 < y < 3.79999999999999994e167

   1. Initial program 78.8%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. 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 97.5

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

   4. Applied rewrites 97.5%

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

   if 3.79999999999999994e167 < y

   1. Initial program 66.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+244}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{x - y} \cdot \left(2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-88}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{elif}\;y \leq 0.000125:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e-88)
   (* -2.0 (fma (/ x y) x x))
   (if (<= y 0.000125) (* (fma (/ 2.0 x) y 2.0) y) (* -2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e-88) {
		tmp = -2.0 * fma((x / y), x, x);
	} else if (y <= 0.000125) {
		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 <= -1.65e-88)
		tmp = Float64(-2.0 * fma(Float64(x / y), x, x));
	elseif (y <= 0.000125)
		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, -1.65e-88], N[(-2.0 * N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.000125], N[(N[(N[(2.0 / x), $MachinePrecision] * y + 2.0), $MachinePrecision] * y), $MachinePrecision], N[(-2.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.64999999999999997e-88

   1. Initial program 73.4%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 73.7

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

   5. Applied rewrites 73.7%

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

   if -1.64999999999999997e-88 < y < 1.25e-4

   1. Initial program 77.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

   if 1.25e-4 < y

   1. Initial program 69.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-88}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{elif}\;y \leq 0.000125:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-88}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;y \leq 0.000125:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e-88)
   (* -2.0 x)
   (if (<= y 0.000125) (* (fma (/ 2.0 x) y 2.0) y) (* -2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e-88) {
		tmp = -2.0 * x;
	} else if (y <= 0.000125) {
		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 <= -1.65e-88)
		tmp = Float64(-2.0 * x);
	elseif (y <= 0.000125)
		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, -1.65e-88], N[(-2.0 * x), $MachinePrecision], If[LessEqual[y, 0.000125], 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 < -1.64999999999999997e-88 or 1.25e-4 < y

   1. Initial program 71.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if -1.64999999999999997e-88 < y < 1.25e-4

   1. Initial program 77.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

Alternative 5: 74.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-88}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;y \leq 0.225:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e-88) (* -2.0 x) (if (<= y 0.225) (* 2.0 y) (* -2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e-88) {
		tmp = -2.0 * x;
	} else if (y <= 0.225) {
		tmp = 2.0 * y;
	} 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 <= (-1.65d-88)) then
        tmp = (-2.0d0) * x
    else if (y <= 0.225d0) then
        tmp = 2.0d0 * y
    else
        tmp = (-2.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.65e-88) {
		tmp = -2.0 * x;
	} else if (y <= 0.225) {
		tmp = 2.0 * y;
	} else {
		tmp = -2.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.65e-88:
		tmp = -2.0 * x
	elif y <= 0.225:
		tmp = 2.0 * y
	else:
		tmp = -2.0 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65e-88)
		tmp = Float64(-2.0 * x);
	elseif (y <= 0.225)
		tmp = Float64(2.0 * y);
	else
		tmp = Float64(-2.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.65e-88)
		tmp = -2.0 * x;
	elseif (y <= 0.225)
		tmp = 2.0 * y;
	else
		tmp = -2.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65e-88], N[(-2.0 * x), $MachinePrecision], If[LessEqual[y, 0.225], N[(2.0 * y), $MachinePrecision], N[(-2.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.64999999999999997e-88 or 0.225000000000000006 < y

   1. Initial program 71.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if -1.64999999999999997e-88 < y < 0.225000000000000006

   1. Initial program 77.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 81.0

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

   5. Applied rewrites 81.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-88}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;y \leq 0.225:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.4% 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 74.5%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 51.6

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

5. Applied rewrites 51.6%

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

Developer Target 1: 99.5% 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 2024243 
(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)))