Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.7% → 99.5%

Time: 3.1s

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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+75}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e+75)
   (* y (/ (* x 2.0) (- x y)))
   (if (<= x 3.8e-27)
     (* x (/ (* 2.0 y) (- x y)))
     (/ y (fma (/ y x) -0.5 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.5e+75) {
		tmp = y * ((x * 2.0) / (x - y));
	} else if (x <= 3.8e-27) {
		tmp = x * ((2.0 * y) / (x - y));
	} else {
		tmp = y / fma((y / x), -0.5, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.5e+75)
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	elseif (x <= 3.8e-27)
		tmp = Float64(x * Float64(Float64(2.0 * y) / Float64(x - y)));
	else
		tmp = Float64(y / fma(Float64(y / x), -0.5, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.5e+75], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e-27], N[(x * N[(N[(2.0 * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $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 < -2.5000000000000001e75

   1. Initial program 73.3%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 -2.5000000000000001e75 < x < 3.8e-27

   1. Initial program 82.6%

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

      1. associate-/l* 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 29.6%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{x - y}{y}}\right)\right)} \]

      2. expm1-udef 2.3%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{\frac{x - y}{y}}\right)} - 1\right)} \]

      3. associate-/r/ 2.3%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{x - y} \cdot y}\right)} - 1\right) \]

      4. *-commutative 2.3%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \frac{2}{x - y}}\right)} - 1\right) \]

   5. Applied egg-rr 2.3%

      \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \frac{2}{x - y}\right)} - 1\right)} \]
   6. Step-by-step derivation

      1. expm1-def 29.6%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{2}{x - y}\right)\right)} \]

      2. expm1-log1p 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l/ 100.0%

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

   7. Simplified 100.0%

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

   if 3.8e-27 < x

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+75}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+75} \lor \neg \left(x \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.5e+75) (not (<= x 4e-32)))
   (* y (/ (* x 2.0) (- x y)))
   (* x (/ (* 2.0 y) (- x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.5e+75) || !(x <= 4e-32)) {
		tmp = y * ((x * 2.0) / (x - 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 ((x <= (-2.5d+75)) .or. (.not. (x <= 4d-32))) then
        tmp = y * ((x * 2.0d0) / (x - 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 ((x <= -2.5e+75) || !(x <= 4e-32)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = x * ((2.0 * y) / (x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.5e+75) or not (x <= 4e-32):
		tmp = y * ((x * 2.0) / (x - y))
	else:
		tmp = x * ((2.0 * y) / (x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.5e+75) || !(x <= 4e-32))
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(x * Float64(Float64(2.0 * y) / Float64(x - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.5e+75) || ~((x <= 4e-32)))
		tmp = y * ((x * 2.0) / (x - y));
	else
		tmp = x * ((2.0 * y) / (x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.5e+75], N[Not[LessEqual[x, 4e-32]], $MachinePrecision]], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5000000000000001e75 or 4.00000000000000022e-32 < x

   1. Initial program 71.8%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 -2.5000000000000001e75 < x < 4.00000000000000022e-32

   1. Initial program 82.6%

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

      1. associate-/l* 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 29.6%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{x - y}{y}}\right)\right)} \]

      2. expm1-udef 2.3%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{\frac{x - y}{y}}\right)} - 1\right)} \]

      3. associate-/r/ 2.3%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{x - y} \cdot y}\right)} - 1\right) \]

      4. *-commutative 2.3%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \frac{2}{x - y}}\right)} - 1\right) \]

   5. Applied egg-rr 2.3%

      \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \frac{2}{x - y}\right)} - 1\right)} \]
   6. Step-by-step derivation

      1. expm1-def 29.6%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{2}{x - y}\right)\right)} \]

      2. expm1-log1p 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l/ 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+75} \lor \neg \left(x \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \end{array} \]

Alternative 3: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+75}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+151}:\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e+75)
   (* 2.0 y)
   (if (<= x 1.4e+151) (* x (/ 2.0 (/ (- x y) y))) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.2e+75) {
		tmp = 2.0 * y;
	} else if (x <= 1.4e+151) {
		tmp = x * (2.0 / ((x - y) / y));
	} 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 <= (-3.2d+75)) then
        tmp = 2.0d0 * y
    else if (x <= 1.4d+151) then
        tmp = x * (2.0d0 / ((x - y) / y))
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.2e+75) {
		tmp = 2.0 * y;
	} else if (x <= 1.4e+151) {
		tmp = x * (2.0 / ((x - y) / y));
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.2e+75:
		tmp = 2.0 * y
	elif x <= 1.4e+151:
		tmp = x * (2.0 / ((x - y) / y))
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.2e+75)
		tmp = Float64(2.0 * y);
	elseif (x <= 1.4e+151)
		tmp = Float64(x * Float64(2.0 / Float64(Float64(x - y) / y)));
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.2e+75)
		tmp = 2.0 * y;
	elseif (x <= 1.4e+151)
		tmp = x * (2.0 / ((x - y) / y));
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.2e+75], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 1.4e+151], N[(x * N[(2.0 / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999985e75 or 1.39999999999999994e151 < x

   1. Initial program 69.4%

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

      1. associate-/l* 69.6%

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

      2. associate-*r/ 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in x around inf 89.4%

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

   if -3.19999999999999985e75 < x < 1.39999999999999994e151

   1. Initial program 82.0%

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

      1. associate-/l* 99.4%

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

      2. associate-*r/ 99.3%

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

   3. Simplified 99.3%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+75}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+151}:\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]

Alternative 4: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+151}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e+154)
   (* 2.0 y)
   (if (<= x 1.2e+151) (* x (/ (* 2.0 y) (- x y))) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+154) {
		tmp = 2.0 * y;
	} else if (x <= 1.2e+151) {
		tmp = x * ((2.0 * y) / (x - y));
	} 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 <= (-1.7d+154)) then
        tmp = 2.0d0 * y
    else if (x <= 1.2d+151) then
        tmp = x * ((2.0d0 * y) / (x - y))
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+154) {
		tmp = 2.0 * y;
	} else if (x <= 1.2e+151) {
		tmp = x * ((2.0 * y) / (x - y));
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7e+154:
		tmp = 2.0 * y
	elif x <= 1.2e+151:
		tmp = x * ((2.0 * y) / (x - y))
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e+154)
		tmp = Float64(2.0 * y);
	elseif (x <= 1.2e+151)
		tmp = Float64(x * Float64(Float64(2.0 * y) / Float64(x - y)));
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e+154)
		tmp = 2.0 * y;
	elseif (x <= 1.2e+151)
		tmp = x * ((2.0 * y) / (x - y));
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e+154], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 1.2e+151], N[(x * N[(N[(2.0 * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999987e154 or 1.20000000000000005e151 < x

   1. Initial program 65.0%

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

      1. associate-/l* 66.4%

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

      2. associate-*r/ 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in x around inf 90.7%

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

   if -1.69999999999999987e154 < x < 1.20000000000000005e151

   1. Initial program 82.6%

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

      1. associate-/l* 98.0%

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

      2. associate-*r/ 97.9%

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

   3. Simplified 97.9%

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

      1. expm1-log1p-u 36.7%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{x - y}{y}}\right)\right)} \]

      2. expm1-udef 2.6%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{\frac{x - y}{y}}\right)} - 1\right)} \]

      3. associate-/r/ 2.6%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{x - y} \cdot y}\right)} - 1\right) \]

      4. *-commutative 2.6%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \frac{2}{x - y}}\right)} - 1\right) \]

   5. Applied egg-rr 2.6%

      \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \frac{2}{x - y}\right)} - 1\right)} \]
   6. Step-by-step derivation

      1. expm1-def 36.6%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{2}{x - y}\right)\right)} \]

      2. expm1-log1p 97.7%

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

      3. *-commutative 97.7%

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

      4. associate-*l/ 98.0%

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

   7. Simplified 98.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+151}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]

Alternative 5: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+28) (* x -2.0) (if (<= y 8e+56) (* 2.0 y) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+28) {
		tmp = x * -2.0;
	} else if (y <= 8e+56) {
		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 (y <= (-6.5d+28)) then
        tmp = x * (-2.0d0)
    else if (y <= 8d+56) 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 (y <= -6.5e+28) {
		tmp = x * -2.0;
	} else if (y <= 8e+56) {
		tmp = 2.0 * y;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5e+28:
		tmp = x * -2.0
	elif y <= 8e+56:
		tmp = 2.0 * y
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5e+28)
		tmp = Float64(x * -2.0);
	elseif (y <= 8e+56)
		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 (y <= -6.5e+28)
		tmp = x * -2.0;
	elseif (y <= 8e+56)
		tmp = 2.0 * y;
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5e+28], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 8e+56], N[(2.0 * y), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5000000000000001e28 or 8.00000000000000074e56 < y

   1. Initial program 69.4%

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

      1. associate-/l* 100.0%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 76.1%

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

   if -6.5000000000000001e28 < y < 8.00000000000000074e56

   1. Initial program 83.7%

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

      1. associate-/l* 80.8%

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

      2. associate-*r/ 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in x around inf 75.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 6: 50.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. associate-/l* 89.0%

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

   2. associate-*r/ 88.8%

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

3. Simplified 88.8%

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

4. Taylor expanded in x around inf 54.9%

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

5. Final simplification 54.9%

   \[\leadsto 2 \cdot y \]

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 2023174 
(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)))