Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.0% → 99.2%

Time: 5.7s

Alternatives: 5

Speedup: 0.5×

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: 77.0% 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: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{x}{\frac{x}{y} + -1}\\ t_1 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ t_2 := \frac{x \cdot \left(2 \cdot y\right)}{x - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+75}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ x (+ (/ x y) -1.0))))
        (t_1 (/ (* (* x 2.0) y) (- x y)))
        (t_2 (/ (* x (* 2.0 y)) (- x y))))
   (if (<= t_1 -1e-45)
     t_0
     (if (<= t_1 -5e-305)
       t_2
       (if (<= t_1 0.0)
         (* x (* 2.0 (/ y (- x y))))
         (if (<= t_1 2e+75) t_2 t_0))))))

C

double code(double x, double y) {
	double t_0 = 2.0 * (x / ((x / y) + -1.0));
	double t_1 = ((x * 2.0) * y) / (x - y);
	double t_2 = (x * (2.0 * y)) / (x - y);
	double tmp;
	if (t_1 <= -1e-45) {
		tmp = t_0;
	} else if (t_1 <= -5e-305) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = x * (2.0 * (y / (x - y)));
	} else if (t_1 <= 2e+75) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 * (x / ((x / y) + (-1.0d0)))
    t_1 = ((x * 2.0d0) * y) / (x - y)
    t_2 = (x * (2.0d0 * y)) / (x - y)
    if (t_1 <= (-1d-45)) then
        tmp = t_0
    else if (t_1 <= (-5d-305)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = x * (2.0d0 * (y / (x - y)))
    else if (t_1 <= 2d+75) then
        tmp = t_2
    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) + -1.0));
	double t_1 = ((x * 2.0) * y) / (x - y);
	double t_2 = (x * (2.0 * y)) / (x - y);
	double tmp;
	if (t_1 <= -1e-45) {
		tmp = t_0;
	} else if (t_1 <= -5e-305) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = x * (2.0 * (y / (x - y)));
	} else if (t_1 <= 2e+75) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 * (x / ((x / y) + -1.0))
	t_1 = ((x * 2.0) * y) / (x - y)
	t_2 = (x * (2.0 * y)) / (x - y)
	tmp = 0
	if t_1 <= -1e-45:
		tmp = t_0
	elif t_1 <= -5e-305:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = x * (2.0 * (y / (x - y)))
	elif t_1 <= 2e+75:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 * Float64(x / Float64(Float64(x / y) + -1.0)))
	t_1 = Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
	t_2 = Float64(Float64(x * Float64(2.0 * y)) / Float64(x - y))
	tmp = 0.0
	if (t_1 <= -1e-45)
		tmp = t_0;
	elseif (t_1 <= -5e-305)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(x * Float64(2.0 * Float64(y / Float64(x - y))));
	elseif (t_1 <= 2e+75)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 * (x / ((x / y) + -1.0));
	t_1 = ((x * 2.0) * y) / (x - y);
	t_2 = (x * (2.0 * y)) / (x - y);
	tmp = 0.0;
	if (t_1 <= -1e-45)
		tmp = t_0;
	elseif (t_1 <= -5e-305)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = x * (2.0 * (y / (x - y)));
	elseif (t_1 <= 2e+75)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 * N[(x / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(2.0 * y), $MachinePrecision]), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-45], t$95$0, If[LessEqual[t$95$1, -5e-305], t$95$2, If[LessEqual[t$95$1, 0.0], N[(x * N[(2.0 * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+75], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -9.99999999999999984e-46 or 1.99999999999999985e75 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 48.8%

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

      1. associate-/l* 98.6%

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

      2. associate-*l* 98.7%

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

   3. Simplified 98.7%

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

      1. associate-*r/ 98.7%

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

      2. associate-/l* 50.0%

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

      3. add-log-exp 12.8%

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

      4. *-un-lft-identity 12.8%

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

      5. log-prod 12.8%

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

      6. metadata-eval 12.8%

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

      7. add-log-exp 50.0%

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

      8. associate-/l* 98.7%

         \[\leadsto 0 + \color{blue}{x \cdot \frac{2 \cdot y}{x - y}} \]

      9. associate-*r/ 98.7%

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

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{0 + x \cdot \left(2 \cdot \frac{y}{x - y}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 98.7%

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

      2. associate-*r* 98.6%

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

      3. associate-*r/ 48.8%

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

      4. associate-*l/ 97.4%

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

      5. associate-/r/ 98.7%

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

      6. *-commutative 98.7%

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

      7. associate-/l* 98.7%

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

      8. div-sub 98.8%

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

      9. sub-neg 98.8%

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

      10. *-inverses 98.8%

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

      11. metadata-eval 98.8%

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

   8. Simplified 98.8%

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

   if -9.99999999999999984e-46 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -4.99999999999999985e-305 or 0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 1.99999999999999985e75

   1. Initial program 99.3%

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

      1. associate-*l* 99.3%

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

   3. Simplified 99.3%

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

   if -4.99999999999999985e-305 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 0.0

   1. Initial program 8.2%

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

      1. associate-/l* 99.8%

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

      2. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -1 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot \frac{x}{\frac{x}{y} + -1}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\frac{x \cdot \left(2 \cdot y\right)}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot \left(2 \cdot y\right)}{x - y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{x}{\frac{x}{y} + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-163} \lor \neg \left(y \leq 2.06 \cdot 10^{-140}\right):\\ \;\;\;\;2 \cdot \frac{x}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.5e-163) (not (<= y 2.06e-140)))
   (* 2.0 (/ x (+ (/ x y) -1.0)))
   (* 2.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e-163) || !(y <= 2.06e-140)) {
		tmp = 2.0 * (x / ((x / y) + -1.0));
	} 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 ((y <= (-6.5d-163)) .or. (.not. (y <= 2.06d-140))) then
        tmp = 2.0d0 * (x / ((x / y) + (-1.0d0)))
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e-163) || !(y <= 2.06e-140)) {
		tmp = 2.0 * (x / ((x / y) + -1.0));
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.5e-163) or not (y <= 2.06e-140):
		tmp = 2.0 * (x / ((x / y) + -1.0))
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.5e-163) || !(y <= 2.06e-140))
		tmp = Float64(2.0 * Float64(x / Float64(Float64(x / y) + -1.0)));
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.5e-163) || ~((y <= 2.06e-140)))
		tmp = 2.0 * (x / ((x / y) + -1.0));
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.5e-163], N[Not[LessEqual[y, 2.06e-140]], $MachinePrecision]], N[(2.0 * N[(x / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4999999999999999e-163 or 2.06e-140 < y

   1. Initial program 76.1%

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

      1. associate-/l* 98.3%

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

      2. associate-*l* 98.3%

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

   3. Simplified 98.3%

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

      1. associate-*r/ 98.3%

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

      2. associate-/l* 76.1%

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

      3. add-log-exp 8.4%

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

      4. *-un-lft-identity 8.4%

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

      5. log-prod 8.4%

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

      6. metadata-eval 8.4%

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

      7. add-log-exp 76.1%

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

      8. associate-/l* 98.3%

         \[\leadsto 0 + \color{blue}{x \cdot \frac{2 \cdot y}{x - y}} \]

      9. associate-*r/ 98.3%

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

   6. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{0 + x \cdot \left(2 \cdot \frac{y}{x - y}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 98.3%

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

      2. associate-*r* 98.3%

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

      3. associate-*r/ 76.1%

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

      4. associate-*l/ 86.1%

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

      5. associate-/r/ 98.0%

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

      6. *-commutative 98.0%

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

      7. associate-/l* 98.0%

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

      8. div-sub 98.0%

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

      9. sub-neg 98.0%

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

      10. *-inverses 98.0%

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

      11. metadata-eval 98.0%

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

   8. Simplified 98.0%

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

   if -6.4999999999999999e-163 < y < 2.06e-140

   1. Initial program 69.7%

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

      1. associate-/l* 62.5%

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

      2. associate-*l* 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in x around inf 87.8%

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

      1. *-commutative 87.8%

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

   7. Simplified 87.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-163} \lor \neg \left(y \leq 2.06 \cdot 10^{-140}\right):\\ \;\;\;\;2 \cdot \frac{x}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \frac{x}{\frac{x}{y} + -1}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e-162)
   (* 2.0 (/ x (+ (/ x y) -1.0)))
   (if (<= y 2.1e-145) (* 2.0 y) (* x (* 2.0 (/ y (- x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e-162) {
		tmp = 2.0 * (x / ((x / y) + -1.0));
	} else if (y <= 2.1e-145) {
		tmp = 2.0 * y;
	} else {
		tmp = x * (2.0 * (y / (x - y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.05d-162)) then
        tmp = 2.0d0 * (x / ((x / y) + (-1.0d0)))
    else if (y <= 2.1d-145) then
        tmp = 2.0d0 * y
    else
        tmp = x * (2.0d0 * (y / (x - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.05e-162) {
		tmp = 2.0 * (x / ((x / y) + -1.0));
	} else if (y <= 2.1e-145) {
		tmp = 2.0 * y;
	} else {
		tmp = x * (2.0 * (y / (x - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05e-162:
		tmp = 2.0 * (x / ((x / y) + -1.0))
	elif y <= 2.1e-145:
		tmp = 2.0 * y
	else:
		tmp = x * (2.0 * (y / (x - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e-162)
		tmp = Float64(2.0 * Float64(x / Float64(Float64(x / y) + -1.0)));
	elseif (y <= 2.1e-145)
		tmp = Float64(2.0 * y);
	else
		tmp = Float64(x * Float64(2.0 * Float64(y / Float64(x - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.05e-162)
		tmp = 2.0 * (x / ((x / y) + -1.0));
	elseif (y <= 2.1e-145)
		tmp = 2.0 * y;
	else
		tmp = x * (2.0 * (y / (x - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e-162], N[(2.0 * N[(x / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-145], N[(2.0 * y), $MachinePrecision], N[(x * N[(2.0 * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e-162

   1. Initial program 78.6%

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

      1. associate-/l* 98.9%

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

      2. associate-*l* 98.9%

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

   3. Simplified 98.9%

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

      1. associate-*r/ 98.9%

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

      2. associate-/l* 78.6%

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

      3. add-log-exp 11.3%

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

      4. *-un-lft-identity 11.3%

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

      5. log-prod 11.3%

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

      6. metadata-eval 11.3%

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

      7. add-log-exp 78.6%

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

      8. associate-/l* 98.9%

         \[\leadsto 0 + \color{blue}{x \cdot \frac{2 \cdot y}{x - y}} \]

      9. associate-*r/ 98.9%

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

   6. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{0 + x \cdot \left(2 \cdot \frac{y}{x - y}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 98.9%

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

      2. associate-*r* 98.9%

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

      3. associate-*r/ 78.6%

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

      4. associate-*l/ 87.6%

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

      5. associate-/r/ 99.0%

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

      6. *-commutative 99.0%

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

      7. associate-/l* 99.0%

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

      8. div-sub 99.1%

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

      9. sub-neg 99.1%

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

      10. *-inverses 99.1%

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

      11. metadata-eval 99.1%

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

   8. Simplified 99.1%

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

   if -1.05e-162 < y < 2.09999999999999991e-145

   1. Initial program 69.7%

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

      1. associate-/l* 62.5%

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

      2. associate-*l* 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in x around inf 87.8%

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

      1. *-commutative 87.8%

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

   7. Simplified 87.8%

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

   if 2.09999999999999991e-145 < y

   1. Initial program 73.3%

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

      1. associate-/l* 97.7%

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

      2. associate-*l* 97.7%

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

   3. Simplified 97.7%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \frac{x}{\frac{x}{y} + -1}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.56 \cdot 10^{-12} \lor \neg \left(x \leq 3.9 \cdot 10^{+133}\right):\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.56e-12) (not (<= x 3.9e+133))) (* 2.0 y) (* x -2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.56e-12) || !(x <= 3.9e+133)) {
		tmp = 2.0 * y;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.56d-12)) .or. (.not. (x <= 3.9d+133))) then
        tmp = 2.0d0 * y
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.56e-12) || !(x <= 3.9e+133)) {
		tmp = 2.0 * y;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.56e-12) or not (x <= 3.9e+133):
		tmp = 2.0 * y
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.56e-12) || !(x <= 3.9e+133))
		tmp = Float64(2.0 * y);
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.56e-12) || ~((x <= 3.9e+133)))
		tmp = 2.0 * y;
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.56e-12], N[Not[LessEqual[x, 3.9e+133]], $MachinePrecision]], N[(2.0 * y), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.56000000000000002e-12 or 3.90000000000000014e133 < x

   1. Initial program 68.6%

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

      1. associate-/l* 76.7%

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

      2. associate-*l* 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x around inf 77.6%

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

      1. *-commutative 77.6%

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

   7. Simplified 77.6%

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

   if -1.56000000000000002e-12 < x < 3.90000000000000014e133

   1. Initial program 78.9%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 74.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.56 \cdot 10^{-12} \lor \neg \left(x \leq 3.9 \cdot 10^{+133}\right):\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. associate-/l* 89.9%

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

   2. associate-*l* 90.0%

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

3. Simplified 90.0%

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

5. Taylor expanded in x around 0 52.9%

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

6. Final simplification 52.9%

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

Developer target: 99.4% accurate, 0.5× 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 2024067 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :alt
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

  (/ (* (* x 2.0) y) (- x y)))