Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.7% → 99.8%

Time: 7.5s

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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot x\right) \cdot \frac{y}{x - y}\\ \mathbf{if}\;y \leq -6 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{-2}{y - x} \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 x) (/ y (- x y)))))
   (if (<= y -6e-28)
     t_0
     (if (<= y 4.9e-11) (* (* (/ -2.0 (- y x)) x) y) t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 * x) * (y / (x - y));
	double tmp;
	if (y <= -6e-28) {
		tmp = t_0;
	} else if (y <= 4.9e-11) {
		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 / (x - y))
    if (y <= (-6d-28)) then
        tmp = t_0
    else if (y <= 4.9d-11) 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 / (x - y));
	double tmp;
	if (y <= -6e-28) {
		tmp = t_0;
	} else if (y <= 4.9e-11) {
		tmp = ((-2.0 / (y - x)) * x) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 * x) * (y / (x - y))
	tmp = 0
	if y <= -6e-28:
		tmp = t_0
	elif y <= 4.9e-11:
		tmp = ((-2.0 / (y - x)) * x) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * x) * Float64(y / Float64(x - y)))
	tmp = 0.0
	if (y <= -6e-28)
		tmp = t_0;
	elseif (y <= 4.9e-11)
		tmp = Float64(Float64(Float64(-2.0 / Float64(y - x)) * x) * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 * x) * (y / (x - y));
	tmp = 0.0;
	if (y <= -6e-28)
		tmp = t_0;
	elseif (y <= 4.9e-11)
		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[(2.0 * x), $MachinePrecision] * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e-28], t$95$0, If[LessEqual[y, 4.9e-11], N[(N[(N[(-2.0 / N[(y - x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.00000000000000005e-28 or 4.8999999999999999e-11 < y

   1. Initial program 78.4%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. 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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   if -6.00000000000000005e-28 < y < 4.8999999999999999e-11

   1. Initial program 76.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 75.7

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

      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 75.7

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

   4. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. frac-2neg N/A

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

      18. lift-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

   6. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-28}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot \frac{y}{x - y}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{-2}{y - x} \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot \frac{y}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-2}{y - x}\\ t_1 := \left(t\_0 \cdot x\right) \cdot y\\ t_2 := \frac{\left(2 \cdot x\right) \cdot y}{x - y}\\ t_3 := \left(x \cdot y\right) \cdot t\_0\\ \mathbf{if}\;t\_2 \leq -1.12 \cdot 10^{-27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-280}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ -2.0 (- y x)))
        (t_1 (* (* t_0 x) y))
        (t_2 (/ (* (* 2.0 x) y) (- x y)))
        (t_3 (* (* x y) t_0)))
   (if (<= t_2 -1.12e-27)
     t_1
     (if (<= t_2 -1e-280)
       t_3
       (if (<= t_2 0.0) t_1 (if (<= t_2 5e-106) t_3 t_1))))))

C

double code(double x, double y) {
	double t_0 = -2.0 / (y - x);
	double t_1 = (t_0 * x) * y;
	double t_2 = ((2.0 * x) * y) / (x - y);
	double t_3 = (x * y) * t_0;
	double tmp;
	if (t_2 <= -1.12e-27) {
		tmp = t_1;
	} else if (t_2 <= -1e-280) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 5e-106) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (-2.0d0) / (y - x)
    t_1 = (t_0 * x) * y
    t_2 = ((2.0d0 * x) * y) / (x - y)
    t_3 = (x * y) * t_0
    if (t_2 <= (-1.12d-27)) then
        tmp = t_1
    else if (t_2 <= (-1d-280)) then
        tmp = t_3
    else if (t_2 <= 0.0d0) then
        tmp = t_1
    else if (t_2 <= 5d-106) then
        tmp = t_3
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -2.0 / (y - x);
	double t_1 = (t_0 * x) * y;
	double t_2 = ((2.0 * x) * y) / (x - y);
	double t_3 = (x * y) * t_0;
	double tmp;
	if (t_2 <= -1.12e-27) {
		tmp = t_1;
	} else if (t_2 <= -1e-280) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 5e-106) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -2.0 / (y - x)
	t_1 = (t_0 * x) * y
	t_2 = ((2.0 * x) * y) / (x - y)
	t_3 = (x * y) * t_0
	tmp = 0
	if t_2 <= -1.12e-27:
		tmp = t_1
	elif t_2 <= -1e-280:
		tmp = t_3
	elif t_2 <= 0.0:
		tmp = t_1
	elif t_2 <= 5e-106:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-2.0 / Float64(y - x))
	t_1 = Float64(Float64(t_0 * x) * y)
	t_2 = Float64(Float64(Float64(2.0 * x) * y) / Float64(x - y))
	t_3 = Float64(Float64(x * y) * t_0)
	tmp = 0.0
	if (t_2 <= -1.12e-27)
		tmp = t_1;
	elseif (t_2 <= -1e-280)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 5e-106)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -2.0 / (y - x);
	t_1 = (t_0 * x) * y;
	t_2 = ((2.0 * x) * y) / (x - y);
	t_3 = (x * y) * t_0;
	tmp = 0.0;
	if (t_2 <= -1.12e-27)
		tmp = t_1;
	elseif (t_2 <= -1e-280)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 5e-106)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-2.0 / N[(y - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * y), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, -1.12e-27], t$95$1, If[LessEqual[t$95$2, -1e-280], t$95$3, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 5e-106], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -1.1199999999999999e-27 or -9.9999999999999996e-281 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 0.0 or 4.99999999999999983e-106 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 55.8%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. 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 97.6

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

      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 97.6

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

   4. Applied rewrites 97.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. frac-2neg N/A

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

      18. lift-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

   6. Applied rewrites 99.6%

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

   if -1.1199999999999999e-27 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -9.9999999999999996e-281 or 0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 4.99999999999999983e-106

   1. Initial program 98.7%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 98.4

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

   4. Applied rewrites 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq -1.12 \cdot 10^{-27}:\\ \;\;\;\;\left(\frac{-2}{y - x} \cdot x\right) \cdot y\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{-2}{y - x}\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;\left(\frac{-2}{y - x} \cdot x\right) \cdot y\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{-2}{y - x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{y - x} \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot y\right) \cdot \frac{-2}{y - x}\\ t_1 := \frac{\left(2 \cdot x\right) \cdot y}{x - y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-295}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-307}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x y) (/ -2.0 (- y x)))) (t_1 (/ (* (* 2.0 x) y) (- x y))))
   (if (<= t_1 (- INFINITY))
     (* 2.0 y)
     (if (<= t_1 -5e-295)
       t_0
       (if (<= t_1 1e-307)
         (* -2.0 x)
         (if (<= t_1 2e+29) t_0 (* (fma (/ x y) x x) -2.0)))))))

C

double code(double x, double y) {
	double t_0 = (x * y) * (-2.0 / (y - x));
	double t_1 = ((2.0 * x) * y) / (x - y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 2.0 * y;
	} else if (t_1 <= -5e-295) {
		tmp = t_0;
	} else if (t_1 <= 1e-307) {
		tmp = -2.0 * x;
	} else if (t_1 <= 2e+29) {
		tmp = t_0;
	} else {
		tmp = fma((x / y), x, x) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * y) * Float64(-2.0 / 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(2.0 * y);
	elseif (t_1 <= -5e-295)
		tmp = t_0;
	elseif (t_1 <= 1e-307)
		tmp = Float64(-2.0 * x);
	elseif (t_1 <= 2e+29)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(x / y), x, x) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] * N[(-2.0 / N[(y - x), $MachinePrecision]), $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[(2.0 * y), $MachinePrecision], If[LessEqual[t$95$1, -5e-295], t$95$0, If[LessEqual[t$95$1, 1e-307], N[(-2.0 * x), $MachinePrecision], If[LessEqual[t$95$1, 2e+29], t$95$0, N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] * -2.0), $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 5.1%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 57.7

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

   5. Applied rewrites 57.7%

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

   if -inf.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -5.00000000000000008e-295 or 9.99999999999999909e-308 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 1.99999999999999983e29

   1. Initial program 99.1%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. *-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 98.8

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

   4. Applied rewrites 98.8%

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

   if -5.00000000000000008e-295 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 9.99999999999999909e-308

   1. Initial program 14.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if 1.99999999999999983e29 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 30.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-2 \cdot x + -2 \cdot \frac{{x}^{2}}{y}} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq -\infty:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq -5 \cdot 10^{-295}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{-2}{y - x}\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq 10^{-307}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{-2}{y - x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e-45)
   (* 2.0 y)
   (if (<= x 1.7e+27) (* (fma (/ x y) x x) -2.0) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e-45) {
		tmp = 2.0 * y;
	} else if (x <= 1.7e+27) {
		tmp = fma((x / y), x, x) * -2.0;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e-45)
		tmp = Float64(2.0 * y);
	elseif (x <= 1.7e+27)
		tmp = Float64(fma(Float64(x / y), x, x) * -2.0);
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e-45], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 1.7e+27], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] * -2.0), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4000000000000001e-45 or 1.7e27 < x

   1. Initial program 75.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -1.4000000000000001e-45 < x < 1.7e27

   1. Initial program 80.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 82.2

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

   5. Applied rewrites 82.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e-45) (* 2.0 y) (if (<= x 1.7e+27) (* -2.0 x) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e-45) {
		tmp = 2.0 * y;
	} else if (x <= 1.7e+27) {
		tmp = -2.0 * x;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d-45)) then
        tmp = 2.0d0 * y
    else if (x <= 1.7d+27) then
        tmp = (-2.0d0) * x
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2e-45) {
		tmp = 2.0 * y;
	} else if (x <= 1.7e+27) {
		tmp = -2.0 * x;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e-45:
		tmp = 2.0 * y
	elif x <= 1.7e+27:
		tmp = -2.0 * x
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e-45)
		tmp = Float64(2.0 * y);
	elseif (x <= 1.7e+27)
		tmp = Float64(-2.0 * x);
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e-45)
		tmp = 2.0 * y;
	elseif (x <= 1.7e+27)
		tmp = -2.0 * x;
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e-45], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 1.7e+27], N[(-2.0 * x), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999997e-45 or 1.7e27 < x

   1. Initial program 75.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -1.99999999999999997e-45 < x < 1.7e27

   1. Initial program 80.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
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 77.7%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 54.8

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

5. Applied rewrites 54.8%

   \[\leadsto \color{blue}{-2 \cdot x} \]
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 2024277 
(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)))