Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.2% → 99.7%

Time: 4.9s

Alternatives: 7

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: 76.2% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-64} \lor \neg \left(y \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1e-64) (not (<= y 2e-7)))
   (/ (* x 2.0) (+ (/ x y) -1.0))
   (/ y (fma (/ y x) -0.5 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1e-64) || !(y <= 2e-7)) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else {
		tmp = y / fma((y / x), -0.5, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1e-64) || !(y <= 2e-7))
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0));
	else
		tmp = Float64(y / fma(Float64(y / x), -0.5, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1e-64], N[Not[LessEqual[y, 2e-7]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(y / x), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999965e-65 or 1.9999999999999999e-7 < y

   1. Initial program 83.4%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 100.0%

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

   if -9.99999999999999965e-65 < y < 1.9999999999999999e-7

   1. Initial program 69.7%

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

      1. *-commutative 69.7%

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

      2. associate-/l* 99.1%

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

      3. div-sub 99.1%

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

      4. sub-neg 99.1%

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

      5. +-commutative 99.1%

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

      6. distribute-neg-frac 99.1%

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

      7. neg-mul-1 99.1%

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

      8. *-commutative 99.1%

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

      9. times-frac 99.1%

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

      10. metadata-eval 99.1%

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

      11. metadata-eval 99.1%

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

      12. metadata-eval 99.1%

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

      13. *-inverses 99.1%

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

      14. associate-/r* 99.1%

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

      15. *-commutative 99.1%

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

      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 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-64} \lor \neg \left(y \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \]

Alternative 2: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e-108)
   (/ x (fma 0.5 (/ x y) -0.5))
   (if (<= y 9.2e-89)
     (* y (/ (* x 2.0) (- x y)))
     (/ (* x 2.0) (+ (/ x y) -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e-108) {
		tmp = x / fma(0.5, (x / y), -0.5);
	} else if (y <= 9.2e-89) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e-108)
		tmp = Float64(x / fma(0.5, Float64(x / y), -0.5));
	elseif (y <= 9.2e-89)
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e-108], N[(x / N[(0.5 * N[(x / y), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e-89], 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] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.99999999999999941e-108

   1. Initial program 80.9%

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

      1. associate-*l* 82.0%

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

      2. associate-/l* 98.9%

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

      3. div-sub 98.9%

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

      4. remove-double-neg 98.9%

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

      5. remove-double-neg 98.9%

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

      6. *-lft-identity 98.9%

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

      7. *-inverses 98.9%

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

      8. times-frac 98.9%

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

      9. *-inverses 98.9%

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

      10. *-inverses 98.9%

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

      11. associate-/r* 98.9%

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

      12. *-commutative 98.9%

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

      13. fma-neg 98.9%

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

      14. *-commutative 98.9%

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

      15. associate-/r* 98.9%

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

      16. *-inverses 98.9%

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

      17. metadata-eval 98.9%

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

      18. *-commutative 98.9%

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

      19. associate-/r* 98.9%

          \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\color{blue}{\frac{\frac{y}{y}}{2}}\right)} \]

      20. *-inverses 98.9%

          \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\frac{\color{blue}{1}}{2}\right)} \]

      21. metadata-eval 98.9%

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

      22. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   if -8.99999999999999941e-108 < y < 9.200000000000001e-89

   1. Initial program 65.6%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 9.200000000000001e-89 < y

   1. Initial program 85.1%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{x \cdot 2}{\color{blue}{\frac{x}{y} - 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \end{array} \]

Alternative 3: 87.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{y \cdot x}{x - y}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-172}:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{elif}\;x \leq 1.78 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ (* y x) (- x y)))))
   (if (<= x -3.2e+145)
     (/ y 0.5)
     (if (<= x -7.6e-159)
       t_0
       (if (<= x 9.2e-172) (/ x -0.5) (if (<= x 1.78e+134) t_0 (/ y 0.5)))))))

C

double code(double x, double y) {
	double t_0 = 2.0 * ((y * x) / (x - y));
	double tmp;
	if (x <= -3.2e+145) {
		tmp = y / 0.5;
	} else if (x <= -7.6e-159) {
		tmp = t_0;
	} else if (x <= 9.2e-172) {
		tmp = x / -0.5;
	} else if (x <= 1.78e+134) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * ((y * x) / (x - y))
    if (x <= (-3.2d+145)) then
        tmp = y / 0.5d0
    else if (x <= (-7.6d-159)) then
        tmp = t_0
    else if (x <= 9.2d-172) then
        tmp = x / (-0.5d0)
    else if (x <= 1.78d+134) then
        tmp = t_0
    else
        tmp = y / 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 * ((y * x) / (x - y));
	double tmp;
	if (x <= -3.2e+145) {
		tmp = y / 0.5;
	} else if (x <= -7.6e-159) {
		tmp = t_0;
	} else if (x <= 9.2e-172) {
		tmp = x / -0.5;
	} else if (x <= 1.78e+134) {
		tmp = t_0;
	} else {
		tmp = y / 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 * ((y * x) / (x - y))
	tmp = 0
	if x <= -3.2e+145:
		tmp = y / 0.5
	elif x <= -7.6e-159:
		tmp = t_0
	elif x <= 9.2e-172:
		tmp = x / -0.5
	elif x <= 1.78e+134:
		tmp = t_0
	else:
		tmp = y / 0.5
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 * Float64(Float64(y * x) / Float64(x - y)))
	tmp = 0.0
	if (x <= -3.2e+145)
		tmp = Float64(y / 0.5);
	elseif (x <= -7.6e-159)
		tmp = t_0;
	elseif (x <= 9.2e-172)
		tmp = Float64(x / -0.5);
	elseif (x <= 1.78e+134)
		tmp = t_0;
	else
		tmp = Float64(y / 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 * ((y * x) / (x - y));
	tmp = 0.0;
	if (x <= -3.2e+145)
		tmp = y / 0.5;
	elseif (x <= -7.6e-159)
		tmp = t_0;
	elseif (x <= 9.2e-172)
		tmp = x / -0.5;
	elseif (x <= 1.78e+134)
		tmp = t_0;
	else
		tmp = y / 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 * N[(N[(y * x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e+145], N[(y / 0.5), $MachinePrecision], If[LessEqual[x, -7.6e-159], t$95$0, If[LessEqual[x, 9.2e-172], N[(x / -0.5), $MachinePrecision], If[LessEqual[x, 1.78e+134], t$95$0, N[(y / 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.20000000000000008e145 or 1.78000000000000005e134 < x

   1. Initial program 65.2%

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

      1. *-commutative 65.2%

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

      2. associate-/l* 98.4%

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

      3. div-sub 98.4%

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

      4. sub-neg 98.4%

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

      5. +-commutative 98.4%

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

      6. distribute-neg-frac 98.4%

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

      7. neg-mul-1 98.4%

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

      8. *-commutative 98.4%

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

      9. times-frac 98.4%

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

      10. metadata-eval 98.4%

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

      11. metadata-eval 98.4%

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

      12. metadata-eval 98.4%

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

      13. *-inverses 98.4%

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

      14. associate-/r* 98.4%

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

      15. *-commutative 98.4%

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

      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%

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

   4. Taylor expanded in y around 0 86.6%

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

   if -3.20000000000000008e145 < x < -7.6000000000000002e-159 or 9.1999999999999998e-172 < x < 1.78000000000000005e134

   1. Initial program 93.6%

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

      1. *-commutative 93.6%

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

      2. associate-*l* 93.6%

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

      3. associate-*r/ 93.6%

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

   3. Simplified 93.6%

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

   if -7.6000000000000002e-159 < x < 9.1999999999999998e-172

   1. Initial program 61.3%

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

      1. associate-*l* 61.3%

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

      2. associate-/l* 100.0%

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

      3. div-sub 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. *-lft-identity 100.0%

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

      7. *-inverses 100.0%

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

      8. times-frac 100.0%

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

      9. *-inverses 100.0%

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

      10. *-inverses 100.0%

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

      11. associate-/r* 100.0%

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

      12. *-commutative 100.0%

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

      13. fma-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-/r* 100.0%

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

      16. *-inverses 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-commutative 100.0%

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

      19. associate-/r* 100.0%

          \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\color{blue}{\frac{\frac{y}{y}}{2}}\right)} \]

      20. *-inverses 100.0%

          \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\frac{\color{blue}{1}}{2}\right)} \]

      21. metadata-eval 100.0%

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

      22. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 86.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-159}:\\ \;\;\;\;2 \cdot \frac{y \cdot x}{x - y}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-172}:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{elif}\;x \leq 1.78 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \frac{y \cdot x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-64} \lor \neg \left(y \leq 0.0001\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x - y}{x \cdot 2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1e-64) (not (<= y 0.0001)))
   (/ (* x 2.0) (+ (/ x y) -1.0))
   (/ y (/ (- x y) (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1e-64) || !(y <= 0.0001)) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else {
		tmp = y / ((x - y) / (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 <= (-1d-64)) .or. (.not. (y <= 0.0001d0))) then
        tmp = (x * 2.0d0) / ((x / y) + (-1.0d0))
    else
        tmp = y / ((x - y) / (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1e-64) || !(y <= 0.0001)) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else {
		tmp = y / ((x - y) / (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1e-64) or not (y <= 0.0001):
		tmp = (x * 2.0) / ((x / y) + -1.0)
	else:
		tmp = y / ((x - y) / (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1e-64) || !(y <= 0.0001))
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0));
	else
		tmp = Float64(y / Float64(Float64(x - y) / Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1e-64) || ~((y <= 0.0001)))
		tmp = (x * 2.0) / ((x / y) + -1.0);
	else
		tmp = y / ((x - y) / (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1e-64], N[Not[LessEqual[y, 0.0001]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(x - y), $MachinePrecision] / N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999965e-65 or 1.00000000000000005e-4 < y

   1. Initial program 83.4%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 100.0%

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

   if -9.99999999999999965e-65 < y < 1.00000000000000005e-4

   1. Initial program 69.7%

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

      1. associate-*l/ 99.0%

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

   3. Simplified 99.0%

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

      1. *-commutative 99.0%

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

      2. clear-num 99.0%

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

      3. un-div-inv 99.1%

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

   5. Applied egg-rr 99.1%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-64} \lor \neg \left(y \leq 0.0001\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x - y}{x \cdot 2}}\\ \end{array} \]

Alternative 5: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+202}:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+143}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e+202)
   (/ x -0.5)
   (if (<= y 2.5e+143) (* y (/ (* x 2.0) (- x y))) (/ x -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e+202) {
		tmp = x / -0.5;
	} else if (y <= 2.5e+143) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = x / -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 (y <= (-7d+202)) then
        tmp = x / (-0.5d0)
    else if (y <= 2.5d+143) then
        tmp = y * ((x * 2.0d0) / (x - y))
    else
        tmp = x / (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7e+202) {
		tmp = x / -0.5;
	} else if (y <= 2.5e+143) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = x / -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e+202:
		tmp = x / -0.5
	elif y <= 2.5e+143:
		tmp = y * ((x * 2.0) / (x - y))
	else:
		tmp = x / -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e+202)
		tmp = Float64(x / -0.5);
	elseif (y <= 2.5e+143)
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(x / -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7e+202)
		tmp = x / -0.5;
	elseif (y <= 2.5e+143)
		tmp = y * ((x * 2.0) / (x - y));
	else
		tmp = x / -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e+202], N[(x / -0.5), $MachinePrecision], If[LessEqual[y, 2.5e+143], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999975e202 or 2.50000000000000006e143 < y

   1. Initial program 77.2%

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

      1. associate-*l* 77.2%

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

      2. associate-/l* 99.9%

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

      3. div-sub 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. *-lft-identity 100.0%

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

      7. *-inverses 100.0%

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

      8. times-frac 100.0%

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

      9. *-inverses 100.0%

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

      10. *-inverses 100.0%

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

      11. associate-/r* 100.0%

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

      12. *-commutative 100.0%

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

      13. fma-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-/r* 100.0%

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

      16. *-inverses 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-commutative 100.0%

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

      19. associate-/r* 100.0%

          \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\color{blue}{\frac{\frac{y}{y}}{2}}\right)} \]

      20. *-inverses 100.0%

          \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\frac{\color{blue}{1}}{2}\right)} \]

      21. metadata-eval 100.0%

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

      22. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 92.1%

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

   if -6.99999999999999975e202 < y < 2.50000000000000006e143

   1. Initial program 77.5%

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

      1. associate-*l/ 95.9%

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

   3. Simplified 95.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+202}:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+143}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-0.5}\\ \end{array} \]

Alternative 6: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e-5) (/ y 0.5) (if (<= x 9.5e+54) (/ x -0.5) (/ y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e-5) {
		tmp = y / 0.5;
	} else if (x <= 9.5e+54) {
		tmp = x / -0.5;
	} 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 <= (-1.45d-5)) then
        tmp = y / 0.5d0
    else if (x <= 9.5d+54) then
        tmp = x / (-0.5d0)
    else
        tmp = y / 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.45e-5) {
		tmp = y / 0.5;
	} else if (x <= 9.5e+54) {
		tmp = x / -0.5;
	} else {
		tmp = y / 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.45e-5:
		tmp = y / 0.5
	elif x <= 9.5e+54:
		tmp = x / -0.5
	else:
		tmp = y / 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e-5)
		tmp = Float64(y / 0.5);
	elseif (x <= 9.5e+54)
		tmp = Float64(x / -0.5);
	else
		tmp = Float64(y / 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.45e-5)
		tmp = y / 0.5;
	elseif (x <= 9.5e+54)
		tmp = x / -0.5;
	else
		tmp = y / 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e-5], N[(y / 0.5), $MachinePrecision], If[LessEqual[x, 9.5e+54], N[(x / -0.5), $MachinePrecision], N[(y / 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e-5 or 9.4999999999999999e54 < x

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-/l* 99.0%

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

      3. div-sub 99.0%

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

      4. sub-neg 99.0%

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

      5. +-commutative 99.0%

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

      6. distribute-neg-frac 99.0%

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

      7. neg-mul-1 99.0%

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

      8. *-commutative 99.0%

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

      9. times-frac 99.0%

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

      10. metadata-eval 99.0%

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

      11. metadata-eval 99.0%

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

      12. metadata-eval 99.0%

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

      13. *-inverses 99.0%

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

      14. associate-/r* 99.0%

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

      15. *-commutative 99.0%

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

      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%

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

   4. Taylor expanded in y around 0 78.5%

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

   if -1.45e-5 < x < 9.4999999999999999e54

   1. Initial program 78.5%

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

      1. associate-*l* 78.5%

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

      2. associate-/l* 99.4%

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

      3. div-sub 99.4%

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

      4. remove-double-neg 99.4%

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

      5. remove-double-neg 99.4%

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

      6. *-lft-identity 99.4%

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

      7. *-inverses 99.4%

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

      8. times-frac 99.4%

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

      9. *-inverses 99.4%

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

      10. *-inverses 99.4%

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

      11. associate-/r* 99.4%

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

      12. *-commutative 99.4%

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

      13. fma-neg 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. *-inverses 99.4%

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

      17. metadata-eval 99.4%

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

      18. *-commutative 99.4%

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

      19. associate-/r* 99.4%

          \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\color{blue}{\frac{\frac{y}{y}}{2}}\right)} \]

      20. *-inverses 99.4%

          \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\frac{\color{blue}{1}}{2}\right)} \]

      21. metadata-eval 99.4%

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

      22. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 76.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5}\\ \end{array} \]

Alternative 7: 50.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{-0.5} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x -0.5))

C

double code(double x, double y) {
	return x / -0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (-0.5d0)
end function

Java

public static double code(double x, double y) {
	return x / -0.5;
}

Python

def code(x, y):
	return x / -0.5

Julia

function code(x, y)
	return Float64(x / -0.5)
end

MATLAB

function tmp = code(x, y)
	tmp = x / -0.5;
end

Wolfram

code[x_, y_] := N[(x / -0.5), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. associate-*l* 77.8%

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

   2. associate-/l* 89.4%

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

   3. div-sub 89.4%

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

   4. remove-double-neg 89.4%

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

   5. remove-double-neg 89.4%

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

   6. *-lft-identity 89.4%

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

   7. *-inverses 89.4%

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

   8. times-frac 89.4%

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

   9. *-inverses 89.4%

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

   10. *-inverses 89.4%

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

   11. associate-/r* 89.4%

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

   12. *-commutative 89.4%

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

   13. fma-neg 89.4%

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

   14. *-commutative 89.4%

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

   15. associate-/r* 89.4%

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

   16. *-inverses 89.4%

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

   17. metadata-eval 89.4%

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

   18. *-commutative 89.4%

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

   19. associate-/r* 89.4%

       \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\color{blue}{\frac{\frac{y}{y}}{2}}\right)} \]

   20. *-inverses 89.4%

       \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\frac{\color{blue}{1}}{2}\right)} \]

   21. metadata-eval 89.4%

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

   22. metadata-eval 89.4%

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

3. Simplified 89.4%

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

4. Taylor expanded in x around 0 54.7%

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

5. Final simplification 54.7%

   \[\leadsto \frac{x}{-0.5} \]

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 2023238 
(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)))