Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.4% → 99.8%

Time: 3.7s

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: 76.4% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-55} \lor \neg \left(x \leq 480000\right):\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.45e-55) (not (<= x 480000.0)))
   (/ y (fma (/ y x) -0.5 0.5))
   (/ (* x 2.0) (/ (- x y) y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.45e-55) || !(x <= 480000.0)) {
		tmp = y / fma((y / x), -0.5, 0.5);
	} else {
		tmp = (x * 2.0) / ((x - y) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.45e-55) || !(x <= 480000.0))
		tmp = Float64(y / fma(Float64(y / x), -0.5, 0.5));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.45e-55], N[Not[LessEqual[x, 480000.0]], $MachinePrecision]], N[(y / N[(N[(y / x), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e-55 or 4.8e5 < x

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-/l* 99.2%

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

      3. div-sub 99.2%

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

      4. sub-neg 99.2%

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

      5. +-commutative 99.2%

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

      6. distribute-neg-frac 99.2%

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

      7. neg-mul-1 99.2%

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

      8. *-commutative 99.2%

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

      9. times-frac 99.2%

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

      10. metadata-eval 99.2%

          \[\leadsto \frac{y}{\frac{y}{x} \cdot \color{blue}{-0.5} + \frac{x}{x \cdot 2}} \]

      11. metadata-eval 99.2%

          \[\leadsto \frac{y}{\frac{y}{x} \cdot \color{blue}{\left(-0.5\right)} + \frac{x}{x \cdot 2}} \]

      12. metadata-eval 99.2%

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

      13. *-inverses 99.2%

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

      14. associate-/r* 99.2%

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

      15. *-commutative 99.2%

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

      16. associate-/r* 99.9%

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

      17. *-inverses 99.9%

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

      18. *-inverses 99.9%

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

      19. associate-/r* 99.9%

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

      20. *-commutative 99.9%

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

      21. fma-def 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}} \]

   if -1.45e-55 < x < 4.8e5

   1. Initial program 76.2%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-55} \lor \neg \left(x \leq 480000\right):\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-42} \lor \neg \left(y \leq 3.7 \cdot 10^{+117}\right):\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.4e-42) (not (<= y 3.7e+117)))
   (* x (/ 2.0 (/ (- x y) y)))
   (* y (/ (* x 2.0) (- x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.4e-42) || !(y <= 3.7e+117)) {
		tmp = x * (2.0 / ((x - y) / y));
	} else {
		tmp = y * ((x * 2.0) / (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 <= (-4.4d-42)) .or. (.not. (y <= 3.7d+117))) then
        tmp = x * (2.0d0 / ((x - y) / y))
    else
        tmp = y * ((x * 2.0d0) / (x - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.4e-42) || !(y <= 3.7e+117)) {
		tmp = x * (2.0 / ((x - y) / y));
	} else {
		tmp = y * ((x * 2.0) / (x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.4e-42) or not (y <= 3.7e+117):
		tmp = x * (2.0 / ((x - y) / y))
	else:
		tmp = y * ((x * 2.0) / (x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.4e-42) || !(y <= 3.7e+117))
		tmp = Float64(x * Float64(2.0 / Float64(Float64(x - y) / y)));
	else
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.4e-42) || ~((y <= 3.7e+117)))
		tmp = x * (2.0 / ((x - y) / y));
	else
		tmp = y * ((x * 2.0) / (x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.4e-42], N[Not[LessEqual[y, 3.7e+117]], $MachinePrecision]], N[(x * N[(2.0 / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000001e-42 or 3.6999999999999999e117 < y

   1. Initial program 68.5%

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

      1. associate-/l* 99.1%

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

      2. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   if -4.4000000000000001e-42 < y < 3.6999999999999999e117

   1. Initial program 77.3%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-42} \lor \neg \left(y \leq 3.7 \cdot 10^{+117}\right):\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

Alternative 3: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+176}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+191}:\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e+176)
   (/ y 0.5)
   (if (<= x 2.55e+191) (* x (/ 2.0 (/ (- x y) y))) (/ y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e+176) {
		tmp = y / 0.5;
	} else if (x <= 2.55e+191) {
		tmp = x * (2.0 / ((x - y) / y));
	} else {
		tmp = y / 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9d+176)) then
        tmp = y / 0.5d0
    else if (x <= 2.55d+191) then
        tmp = x * (2.0d0 / ((x - y) / y))
    else
        tmp = y / 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9e+176) {
		tmp = y / 0.5;
	} else if (x <= 2.55e+191) {
		tmp = x * (2.0 / ((x - y) / y));
	} else {
		tmp = y / 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9e+176:
		tmp = y / 0.5
	elif x <= 2.55e+191:
		tmp = x * (2.0 / ((x - y) / y))
	else:
		tmp = y / 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e+176)
		tmp = Float64(y / 0.5);
	elseif (x <= 2.55e+191)
		tmp = Float64(x * Float64(2.0 / Float64(Float64(x - y) / y)));
	else
		tmp = Float64(y / 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e+176)
		tmp = y / 0.5;
	elseif (x <= 2.55e+191)
		tmp = x * (2.0 / ((x - y) / y));
	else
		tmp = y / 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e+176], N[(y / 0.5), $MachinePrecision], If[LessEqual[x, 2.55e+191], N[(x * N[(2.0 / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000007e176 or 2.54999999999999991e191 < x

   1. Initial program 57.9%

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

      1. *-commutative 57.9%

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

      2. associate-/l* 98.1%

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

      3. div-sub 98.1%

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

      4. sub-neg 98.1%

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

      5. +-commutative 98.1%

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

      6. distribute-neg-frac 98.1%

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

      7. neg-mul-1 98.1%

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

      8. *-commutative 98.1%

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

      9. times-frac 98.0%

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

      10. metadata-eval 98.0%

          \[\leadsto \frac{y}{\frac{y}{x} \cdot \color{blue}{-0.5} + \frac{x}{x \cdot 2}} \]

      11. metadata-eval 98.0%

          \[\leadsto \frac{y}{\frac{y}{x} \cdot \color{blue}{\left(-0.5\right)} + \frac{x}{x \cdot 2}} \]

      12. metadata-eval 98.0%

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

      13. *-inverses 98.0%

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

      14. associate-/r* 98.0%

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

      15. *-commutative 98.0%

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

      16. associate-/r* 99.9%

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

      17. *-inverses 99.9%

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

      18. *-inverses 99.9%

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

      19. associate-/r* 99.9%

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

      20. *-commutative 99.9%

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

      21. fma-def 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}} \]

   4. Taylor expanded in y around 0 93.6%

      \[\leadsto \frac{y}{\color{blue}{0.5}} \]

   if -9.00000000000000007e176 < x < 2.54999999999999991e191

   1. Initial program 77.4%

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

      1. associate-/l* 96.3%

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

      2. associate-*r/ 96.2%

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

   3. Simplified 96.2%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+176}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+191}:\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;x \leq 1800000:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5 + \frac{-0.5}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+46)
   (* y (/ (* x 2.0) (- x y)))
   (if (<= x 1800000.0)
     (/ (* x 2.0) (/ (- x y) y))
     (/ y (+ 0.5 (/ -0.5 (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+46) {
		tmp = y * ((x * 2.0) / (x - y));
	} else if (x <= 1800000.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = y / (0.5 + (-0.5 / (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 (x <= (-1.35d+46)) then
        tmp = y * ((x * 2.0d0) / (x - y))
    else if (x <= 1800000.0d0) then
        tmp = (x * 2.0d0) / ((x - y) / y)
    else
        tmp = y / (0.5d0 + ((-0.5d0) / (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+46) {
		tmp = y * ((x * 2.0) / (x - y));
	} else if (x <= 1800000.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = y / (0.5 + (-0.5 / (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e+46:
		tmp = y * ((x * 2.0) / (x - y))
	elif x <= 1800000.0:
		tmp = (x * 2.0) / ((x - y) / y)
	else:
		tmp = y / (0.5 + (-0.5 / (x / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+46)
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	elseif (x <= 1800000.0)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	else
		tmp = Float64(y / Float64(0.5 + Float64(-0.5 / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e+46)
		tmp = y * ((x * 2.0) / (x - y));
	elseif (x <= 1800000.0)
		tmp = (x * 2.0) / ((x - y) / y);
	else
		tmp = y / (0.5 + (-0.5 / (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e+46], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1800000.0], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y / N[(0.5 + N[(-0.5 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001e46

   1. Initial program 65.9%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   if -1.3500000000000001e46 < x < 1.8e6

   1. Initial program 79.5%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   if 1.8e6 < x

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-/l* 98.6%

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

      3. div-sub 98.6%

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

      4. sub-neg 98.6%

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

      5. +-commutative 98.6%

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

      6. distribute-neg-frac 98.6%

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

      7. neg-mul-1 98.6%

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

      8. *-commutative 98.6%

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

      9. times-frac 98.6%

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

      10. metadata-eval 98.6%

          \[\leadsto \frac{y}{\frac{y}{x} \cdot \color{blue}{-0.5} + \frac{x}{x \cdot 2}} \]

      11. metadata-eval 98.6%

          \[\leadsto \frac{y}{\frac{y}{x} \cdot \color{blue}{\left(-0.5\right)} + \frac{x}{x \cdot 2}} \]

      12. metadata-eval 98.6%

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

      13. *-inverses 98.6%

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

      14. associate-/r* 98.6%

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

      15. *-commutative 98.6%

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

      16. associate-/r* 99.9%

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

      17. *-inverses 99.9%

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

      18. *-inverses 99.9%

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

      19. associate-/r* 99.9%

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

      20. *-commutative 99.9%

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

      21. fma-def 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}} \]
   4. Step-by-step derivation

      1. fma-udef 99.9%

         \[\leadsto \frac{y}{\color{blue}{\frac{y}{x} \cdot -0.5 + 0.5}} \]

      2. clear-num 99.9%

         \[\leadsto \frac{y}{\color{blue}{\frac{1}{\frac{x}{y}}} \cdot -0.5 + 0.5} \]

      3. associate-*l/ 99.9%

         \[\leadsto \frac{y}{\color{blue}{\frac{1 \cdot -0.5}{\frac{x}{y}}} + 0.5} \]

      4. metadata-eval 99.9%

         \[\leadsto \frac{y}{\frac{\color{blue}{-0.5}}{\frac{x}{y}} + 0.5} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \frac{y}{\color{blue}{\frac{-0.5}{\frac{x}{y}} + 0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;x \leq 1800000:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5 + \frac{-0.5}{\frac{x}{y}}}\\ \end{array} \]

Alternative 5: 74.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-30}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e-30) (* x -2.0) (if (<= y 3.8e+66) (/ y 0.5) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e-30) {
		tmp = x * -2.0;
	} else if (y <= 3.8e+66) {
		tmp = y / 0.5;
	} 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 (y <= (-4d-30)) then
        tmp = x * (-2.0d0)
    else if (y <= 3.8d+66) then
        tmp = y / 0.5d0
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4e-30) {
		tmp = x * -2.0;
	} else if (y <= 3.8e+66) {
		tmp = y / 0.5;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4e-30:
		tmp = x * -2.0
	elif y <= 3.8e+66:
		tmp = y / 0.5
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e-30)
		tmp = Float64(x * -2.0);
	elseif (y <= 3.8e+66)
		tmp = Float64(y / 0.5);
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4e-30)
		tmp = x * -2.0;
	elseif (y <= 3.8e+66)
		tmp = y / 0.5;
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e-30], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 3.8e+66], N[(y / 0.5), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4e-30 or 3.8000000000000002e66 < y

   1. Initial program 69.8%

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

      1. associate-/l* 99.1%

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

      2. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 78.2%

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

   if -4e-30 < y < 3.8000000000000002e66

   1. Initial program 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-/l* 99.9%

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

      3. div-sub 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. neg-mul-1 99.9%

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

      8. *-commutative 99.9%

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

      9. times-frac 99.9%

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

      10. metadata-eval 99.9%

          \[\leadsto \frac{y}{\frac{y}{x} \cdot \color{blue}{-0.5} + \frac{x}{x \cdot 2}} \]

      11. metadata-eval 99.9%

          \[\leadsto \frac{y}{\frac{y}{x} \cdot \color{blue}{\left(-0.5\right)} + \frac{x}{x \cdot 2}} \]

      12. metadata-eval 99.9%

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

      13. *-inverses 99.9%

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

      14. associate-/r* 99.9%

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

      15. *-commutative 99.9%

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

      16. associate-/r* 99.9%

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

      17. *-inverses 99.9%

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

      18. *-inverses 99.9%

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

      19. associate-/r* 99.9%

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

      20. *-commutative 99.9%

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

      21. fma-def 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}} \]

   4. Taylor expanded in y around 0 80.0%

      \[\leadsto \frac{y}{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-30}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 6: 51.1% 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 73.4%

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

   1. associate-/l* 88.9%

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

   2. associate-*r/ 89.1%

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

3. Simplified 89.1%

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

4. Taylor expanded in x around 0 47.9%

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

5. Final simplification 47.9%

   \[\leadsto x \cdot -2 \]

Developer target: 99.5% accurate, 0.7× 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 2023185 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (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)))