Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.8% → 99.5%

Time: 3.4s

Alternatives: 5

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.8% 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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;x \leq 10^{-41}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+49)
   (* y (/ (* x 2.0) (- x y)))
   (if (<= x 1e-41) (/ (* x 2.0) (/ (- x y) y)) (/ y (fma (/ y x) -0.5 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e+49) {
		tmp = y * ((x * 2.0) / (x - y));
	} else if (x <= 1e-41) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = y / fma((y / x), -0.5, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e+49)
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	elseif (x <= 1e-41)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	else
		tmp = Float64(y / fma(Float64(y / x), -0.5, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e+49], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-41], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(y / x), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999989e49

   1. Initial program 72.6%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   if -1.99999999999999989e49 < x < 1.00000000000000001e-41

   1. Initial program 75.8%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 1.00000000000000001e-41 < x

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-/l* 100.0%

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

      3. div-sub 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. neg-mul-1 100.0%

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

      8. *-commutative 100.0%

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

      9. times-frac 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-inverses 100.0%

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

      14. associate-/r* 100.0%

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

      15. *-commutative 100.0%

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

      16. associate-/r* 100.0%

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

      17. *-inverses 100.0%

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

      18. *-inverses 100.0%

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

      19. associate-/r* 100.0%

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

      20. *-commutative 100.0%

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

      21. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;x \leq 10^{-41}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \]

Alternative 2: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-230} \lor \neg \left(x \leq 3.4 \cdot 10^{-186}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.8e-230) (not (<= x 3.4e-186)))
   (* y (/ (* x 2.0) (- x y)))
   (* x -2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.8e-230) || !(x <= 3.4e-186)) {
		tmp = y * ((x * 2.0) / (x - 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 <= (-4.8d-230)) .or. (.not. (x <= 3.4d-186))) then
        tmp = y * ((x * 2.0d0) / (x - 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 <= -4.8e-230) || !(x <= 3.4e-186)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.8e-230) or not (x <= 3.4e-186):
		tmp = y * ((x * 2.0) / (x - y))
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.8e-230) || !(x <= 3.4e-186))
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4.8e-230) || ~((x <= 3.4e-186)))
		tmp = y * ((x * 2.0) / (x - y));
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.8e-230], N[Not[LessEqual[x, 3.4e-186]], $MachinePrecision]], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8000000000000004e-230 or 3.3999999999999999e-186 < x

   1. Initial program 76.7%

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

      1. associate-*l/ 95.6%

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

   3. Simplified 95.6%

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

   if -4.8000000000000004e-230 < x < 3.3999999999999999e-186

   1. Initial program 64.2%

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

      1. associate-*l/ 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in x around 0 97.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-230} \lor \neg \left(x \leq 3.4 \cdot 10^{-186}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+52} \lor \neg \left(x \leq 10^{-41}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.1e+52) (not (<= x 1e-41)))
   (* y (/ (* x 2.0) (- x y)))
   (/ (* x 2.0) (/ (- x y) y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.1e+52) || !(x <= 1e-41)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = (x * 2.0) / ((x - y) / 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 <= (-1.1d+52)) .or. (.not. (x <= 1d-41))) then
        tmp = y * ((x * 2.0d0) / (x - y))
    else
        tmp = (x * 2.0d0) / ((x - y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.1e+52) || !(x <= 1e-41)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = (x * 2.0) / ((x - y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.1e+52) or not (x <= 1e-41):
		tmp = y * ((x * 2.0) / (x - y))
	else:
		tmp = (x * 2.0) / ((x - y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.1e+52) || !(x <= 1e-41))
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.1e+52) || ~((x <= 1e-41)))
		tmp = y * ((x * 2.0) / (x - y));
	else
		tmp = (x * 2.0) / ((x - y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.1e+52], N[Not[LessEqual[x, 1e-41]], $MachinePrecision]], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $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 < -1.1e52 or 1.00000000000000001e-41 < x

   1. Initial program 73.8%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   if -1.1e52 < x < 1.00000000000000001e-41

   1. Initial program 75.8%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+52} \lor \neg \left(x \leq 10^{-41}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \]

Alternative 4: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{elif}\;x \leq 8800000000000:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.9e+14)
   (/ y 0.5)
   (if (<= x 8800000000000.0) (* x -2.0) (/ y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+14) {
		tmp = y / 0.5;
	} else if (x <= 8800000000000.0) {
		tmp = x * -2.0;
	} 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 <= (-4.9d+14)) then
        tmp = y / 0.5d0
    else if (x <= 8800000000000.0d0) then
        tmp = x * (-2.0d0)
    else
        tmp = y / 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+14) {
		tmp = y / 0.5;
	} else if (x <= 8800000000000.0) {
		tmp = x * -2.0;
	} else {
		tmp = y / 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.9e+14:
		tmp = y / 0.5
	elif x <= 8800000000000.0:
		tmp = x * -2.0
	else:
		tmp = y / 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e+14)
		tmp = Float64(y / 0.5);
	elseif (x <= 8800000000000.0)
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(y / 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.9e+14)
		tmp = y / 0.5;
	elseif (x <= 8800000000000.0)
		tmp = x * -2.0;
	else
		tmp = y / 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e+14], N[(y / 0.5), $MachinePrecision], If[LessEqual[x, 8800000000000.0], N[(x * -2.0), $MachinePrecision], N[(y / 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9e14 or 8.8e12 < x

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      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 79.7%

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

   if -4.9e14 < x < 8.8e12

   1. Initial program 76.7%

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

      1. associate-*l/ 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in x around 0 77.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{elif}\;x \leq 8800000000000:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5}\\ \end{array} \]

Alternative 5: 51.2% 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 74.8%

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

   1. associate-*l/ 90.5%

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

3. Simplified 90.5%

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

4. Taylor expanded in x around 0 50.2%

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

5. Final simplification 50.2%

   \[\leadsto x \cdot -2 \]

Developer target: 99.4% 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 2023181 
(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)))