Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4

Percentage Accurate: 99.9% → 100.0%

Time: 5.8s

Alternatives: 9

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (+ (+ (+ x y) y) x) z) x))

C

double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x + y) + y) + x) + z) + x
end function

Java

public static double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}

Python

def code(x, y, z):
	return ((((x + y) + y) + x) + z) + x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x + y) + y) + x) + z) + x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x + y) + y) + x) + z) + x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (+ (+ (+ x y) y) x) z) x))

C

double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x + y) + y) + x) + z) + x
end function

Java

public static double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}

Python

def code(x, y, z):
	return ((((x + y) + y) + x) + z) + x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x + y) + y) + x) + z) + x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x + y) + y) + x) + z) + x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ z + \mathsf{fma}\left(3, x, y \cdot 2\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ z (fma 3.0 x (* y 2.0))))

C

double code(double x, double y, double z) {
	return z + fma(3.0, x, (y * 2.0));
}

Julia

function code(x, y, z)
	return Float64(z + fma(3.0, x, Float64(y * 2.0)))
end

Wolfram

code[x_, y_, z_] := N[(z + N[(3.0 * x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
2. Step-by-step derivation

   1. +-commutative 99.9%

      \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

   2. associate-+l+ 99.9%

      \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

   3. +-commutative 99.9%

      \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

   4. +-commutative 99.9%

      \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

   5. associate-+l+ 99.9%

      \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

   6. associate-+r+ 99.9%

      \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

   7. associate-+r+ 99.9%

      \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

   8. +-commutative 99.9%

      \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

   9. count-2 99.9%

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

   10. distribute-lft1-in 99.9%

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

   11. fma-def 100.0%

       \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

   12. metadata-eval 100.0%

       \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

   13. count-2 100.0%

       \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

   14. *-commutative 100.0%

       \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

4. Final simplification 100.0%

   \[\leadsto z + \mathsf{fma}\left(3, x, y \cdot 2\right) \]

Alternative 2: 53.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+105}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+72}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{+33}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-266}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-234}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+96}:\\ \;\;\;\;3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.9e+105)
   z
   (if (<= z -1.2e+72)
     (* y 2.0)
     (if (<= z -4.7e+33)
       z
       (if (<= z 6.5e-266)
         (* 3.0 x)
         (if (<= z 6.5e-234) (* y 2.0) (if (<= z 6.4e+96) (* 3.0 x) z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+105) {
		tmp = z;
	} else if (z <= -1.2e+72) {
		tmp = y * 2.0;
	} else if (z <= -4.7e+33) {
		tmp = z;
	} else if (z <= 6.5e-266) {
		tmp = 3.0 * x;
	} else if (z <= 6.5e-234) {
		tmp = y * 2.0;
	} else if (z <= 6.4e+96) {
		tmp = 3.0 * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.9d+105)) then
        tmp = z
    else if (z <= (-1.2d+72)) then
        tmp = y * 2.0d0
    else if (z <= (-4.7d+33)) then
        tmp = z
    else if (z <= 6.5d-266) then
        tmp = 3.0d0 * x
    else if (z <= 6.5d-234) then
        tmp = y * 2.0d0
    else if (z <= 6.4d+96) then
        tmp = 3.0d0 * x
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+105) {
		tmp = z;
	} else if (z <= -1.2e+72) {
		tmp = y * 2.0;
	} else if (z <= -4.7e+33) {
		tmp = z;
	} else if (z <= 6.5e-266) {
		tmp = 3.0 * x;
	} else if (z <= 6.5e-234) {
		tmp = y * 2.0;
	} else if (z <= 6.4e+96) {
		tmp = 3.0 * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.9e+105:
		tmp = z
	elif z <= -1.2e+72:
		tmp = y * 2.0
	elif z <= -4.7e+33:
		tmp = z
	elif z <= 6.5e-266:
		tmp = 3.0 * x
	elif z <= 6.5e-234:
		tmp = y * 2.0
	elif z <= 6.4e+96:
		tmp = 3.0 * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.9e+105)
		tmp = z;
	elseif (z <= -1.2e+72)
		tmp = Float64(y * 2.0);
	elseif (z <= -4.7e+33)
		tmp = z;
	elseif (z <= 6.5e-266)
		tmp = Float64(3.0 * x);
	elseif (z <= 6.5e-234)
		tmp = Float64(y * 2.0);
	elseif (z <= 6.4e+96)
		tmp = Float64(3.0 * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.9e+105)
		tmp = z;
	elseif (z <= -1.2e+72)
		tmp = y * 2.0;
	elseif (z <= -4.7e+33)
		tmp = z;
	elseif (z <= 6.5e-266)
		tmp = 3.0 * x;
	elseif (z <= 6.5e-234)
		tmp = y * 2.0;
	elseif (z <= 6.4e+96)
		tmp = 3.0 * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.9e+105], z, If[LessEqual[z, -1.2e+72], N[(y * 2.0), $MachinePrecision], If[LessEqual[z, -4.7e+33], z, If[LessEqual[z, 6.5e-266], N[(3.0 * x), $MachinePrecision], If[LessEqual[z, 6.5e-234], N[(y * 2.0), $MachinePrecision], If[LessEqual[z, 6.4e+96], N[(3.0 * x), $MachinePrecision], z]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e105 or -1.20000000000000005e72 < z < -4.6999999999999998e33 or 6.40000000000000013e96 < z

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.9%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.9%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.9%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.9%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.9%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.9%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.9%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.9%

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

      10. distribute-lft1-in 99.9%

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

      11. fma-def 100.0%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in z around inf 82.8%

      \[\leadsto \color{blue}{z} \]

   if -1.9e105 < z < -1.20000000000000005e72 or 6.50000000000000024e-266 < z < 6.4999999999999994e-234

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 100.0%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 100.0%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 100.0%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 100.0%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 100.0%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 100.0%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 100.0%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 100.0%

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

      10. distribute-lft1-in 100.0%

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

      11. fma-def 100.0%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in y around inf 92.5%

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

   if -4.6999999999999998e33 < z < 6.50000000000000024e-266 or 6.4999999999999994e-234 < z < 6.40000000000000013e96

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.8%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.8%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.8%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.8%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.8%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.8%

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

      10. distribute-lft1-in 99.8%

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

      11. fma-def 99.9%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 99.9%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 99.9%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 99.9%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in x around inf 55.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+105}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+72}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{+33}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-266}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-234}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+96}:\\ \;\;\;\;3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+27}:\\ \;\;\;\;z + 3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e+72)
   (+ z (* y 2.0))
   (if (<= y 4.8e+27) (+ z (* 3.0 x)) (+ x (* 2.0 (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+72) {
		tmp = z + (y * 2.0);
	} else if (y <= 4.8e+27) {
		tmp = z + (3.0 * x);
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.2d+72)) then
        tmp = z + (y * 2.0d0)
    else if (y <= 4.8d+27) then
        tmp = z + (3.0d0 * x)
    else
        tmp = x + (2.0d0 * (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+72) {
		tmp = z + (y * 2.0);
	} else if (y <= 4.8e+27) {
		tmp = z + (3.0 * x);
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.2e+72:
		tmp = z + (y * 2.0)
	elif y <= 4.8e+27:
		tmp = z + (3.0 * x)
	else:
		tmp = x + (2.0 * (x + y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e+72)
		tmp = Float64(z + Float64(y * 2.0));
	elseif (y <= 4.8e+27)
		tmp = Float64(z + Float64(3.0 * x));
	else
		tmp = Float64(x + Float64(2.0 * Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.2e+72)
		tmp = z + (y * 2.0);
	elseif (y <= 4.8e+27)
		tmp = z + (3.0 * x);
	else
		tmp = x + (2.0 * (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.2e+72], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+27], N[(z + N[(3.0 * x), $MachinePrecision]), $MachinePrecision], N[(x + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2000000000000001e72

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.8%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.8%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.8%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.9%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.9%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.9%

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

      10. distribute-lft1-in 99.9%

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

      11. fma-def 100.0%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in x around 0 81.2%

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

      1. +-commutative 81.2%

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

   6. Simplified 81.2%

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

   if -3.2000000000000001e72 < y < 4.79999999999999995e27

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.8%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.8%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.8%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.8%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.8%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.8%

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

      10. distribute-lft1-in 99.8%

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

      11. fma-def 99.9%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 99.9%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 99.9%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 99.9%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in y around 0 92.4%

      \[\leadsto \color{blue}{z + 3 \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 92.4%

         \[\leadsto \color{blue}{3 \cdot x + z} \]

   6. Simplified 92.4%

      \[\leadsto \color{blue}{3 \cdot x + z} \]

   if 4.79999999999999995e27 < y

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(x + y\right) + y\right) + x\right) + \left(z + x\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \left(\left(\color{blue}{\left(y + x\right)} + y\right) + x\right) + \left(z + x\right) \]

      3. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(y + x\right) + \left(y + x\right)\right)} + \left(z + x\right) \]

      4. count-2 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 86.0%

      \[\leadsto \color{blue}{x + 2 \cdot \left(x + y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+27}:\\ \;\;\;\;z + 3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 4: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+71}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+26}:\\ \;\;\;\;x + \left(z + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7e+71)
   (+ z (* y 2.0))
   (if (<= y 1.02e+26) (+ x (+ z (* x 2.0))) (+ x (* 2.0 (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e+71) {
		tmp = z + (y * 2.0);
	} else if (y <= 1.02e+26) {
		tmp = x + (z + (x * 2.0));
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-7d+71)) then
        tmp = z + (y * 2.0d0)
    else if (y <= 1.02d+26) then
        tmp = x + (z + (x * 2.0d0))
    else
        tmp = x + (2.0d0 * (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e+71) {
		tmp = z + (y * 2.0);
	} else if (y <= 1.02e+26) {
		tmp = x + (z + (x * 2.0));
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7e+71:
		tmp = z + (y * 2.0)
	elif y <= 1.02e+26:
		tmp = x + (z + (x * 2.0))
	else:
		tmp = x + (2.0 * (x + y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7e+71)
		tmp = Float64(z + Float64(y * 2.0));
	elseif (y <= 1.02e+26)
		tmp = Float64(x + Float64(z + Float64(x * 2.0)));
	else
		tmp = Float64(x + Float64(2.0 * Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7e+71)
		tmp = z + (y * 2.0);
	elseif (y <= 1.02e+26)
		tmp = x + (z + (x * 2.0));
	else
		tmp = x + (2.0 * (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7e+71], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.02e+26], N[(x + N[(z + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.9999999999999998e71

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.8%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.8%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.8%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.9%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.9%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.9%

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

      10. distribute-lft1-in 99.9%

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

      11. fma-def 100.0%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in x around 0 81.2%

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

      1. +-commutative 81.2%

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

   6. Simplified 81.2%

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

   if -6.9999999999999998e71 < y < 1.0200000000000001e26

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. associate-+l+ 99.8%

         \[\leadsto \color{blue}{\left(\left(\left(x + y\right) + y\right) + x\right) + \left(z + x\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \left(\left(\color{blue}{\left(y + x\right)} + y\right) + x\right) + \left(z + x\right) \]

      3. associate-+l+ 99.9%

         \[\leadsto \color{blue}{\left(\left(y + x\right) + \left(y + x\right)\right)} + \left(z + x\right) \]

      4. count-2 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 92.4%

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

   if 1.0200000000000001e26 < y

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(x + y\right) + y\right) + x\right) + \left(z + x\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \left(\left(\color{blue}{\left(y + x\right)} + y\right) + x\right) + \left(z + x\right) \]

      3. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(y + x\right) + \left(y + x\right)\right)} + \left(z + x\right) \]

      4. count-2 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 86.0%

      \[\leadsto \color{blue}{x + 2 \cdot \left(x + y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+71}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+26}:\\ \;\;\;\;x + \left(z + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 5: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+89} \lor \neg \left(x \leq 3.6 \cdot 10^{+79}\right):\\ \;\;\;\;3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.15e+89) (not (<= x 3.6e+79))) (* 3.0 x) (+ z (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.15e+89) || !(x <= 3.6e+79)) {
		tmp = 3.0 * x;
	} else {
		tmp = z + (y * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.15d+89)) .or. (.not. (x <= 3.6d+79))) then
        tmp = 3.0d0 * x
    else
        tmp = z + (y * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.15e+89) || !(x <= 3.6e+79)) {
		tmp = 3.0 * x;
	} else {
		tmp = z + (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.15e+89) or not (x <= 3.6e+79):
		tmp = 3.0 * x
	else:
		tmp = z + (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.15e+89) || !(x <= 3.6e+79))
		tmp = Float64(3.0 * x);
	else
		tmp = Float64(z + Float64(y * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.15e+89) || ~((x <= 3.6e+79)))
		tmp = 3.0 * x;
	else
		tmp = z + (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.15e+89], N[Not[LessEqual[x, 3.6e+79]], $MachinePrecision]], N[(3.0 * x), $MachinePrecision], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1500000000000001e89 or 3.5999999999999999e79 < x

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.8%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.8%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.8%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.8%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.8%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.8%

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

      10. distribute-lft1-in 99.8%

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

      11. fma-def 99.9%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 99.9%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 99.9%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 99.9%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in x around inf 75.9%

      \[\leadsto \color{blue}{3 \cdot x} \]

   if -2.1500000000000001e89 < x < 3.5999999999999999e79

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.9%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.9%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.9%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.9%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.9%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.9%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.9%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.9%

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

      10. distribute-lft1-in 99.9%

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

      11. fma-def 100.0%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in x around 0 85.5%

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

      1. +-commutative 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+89} \lor \neg \left(x \leq 3.6 \cdot 10^{+79}\right):\\ \;\;\;\;3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]

Alternative 6: 83.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+84} \lor \neg \left(x \leq 3.9 \cdot 10^{-104}\right):\\ \;\;\;\;z + 3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.8e+84) (not (<= x 3.9e-104)))
   (+ z (* 3.0 x))
   (+ z (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+84) || !(x <= 3.9e-104)) {
		tmp = z + (3.0 * x);
	} else {
		tmp = z + (y * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-6.8d+84)) .or. (.not. (x <= 3.9d-104))) then
        tmp = z + (3.0d0 * x)
    else
        tmp = z + (y * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+84) || !(x <= 3.9e-104)) {
		tmp = z + (3.0 * x);
	} else {
		tmp = z + (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.8e+84) or not (x <= 3.9e-104):
		tmp = z + (3.0 * x)
	else:
		tmp = z + (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.8e+84) || !(x <= 3.9e-104))
		tmp = Float64(z + Float64(3.0 * x));
	else
		tmp = Float64(z + Float64(y * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6.8e+84) || ~((x <= 3.9e-104)))
		tmp = z + (3.0 * x);
	else
		tmp = z + (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.8e+84], N[Not[LessEqual[x, 3.9e-104]], $MachinePrecision]], N[(z + N[(3.0 * x), $MachinePrecision]), $MachinePrecision], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.7999999999999996e84 or 3.9000000000000002e-104 < x

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.8%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.8%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.8%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.8%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.8%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.8%

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

      10. distribute-lft1-in 99.8%

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

      11. fma-def 99.9%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 99.9%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 99.9%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 99.9%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in y around 0 85.2%

      \[\leadsto \color{blue}{z + 3 \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 85.2%

         \[\leadsto \color{blue}{3 \cdot x + z} \]

   6. Simplified 85.2%

      \[\leadsto \color{blue}{3 \cdot x + z} \]

   if -6.7999999999999996e84 < x < 3.9000000000000002e-104

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.9%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.9%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.9%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.9%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.9%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.9%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.9%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.9%

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

      10. distribute-lft1-in 99.9%

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

      11. fma-def 100.0%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in x around 0 91.2%

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

      1. +-commutative 91.2%

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

   6. Simplified 91.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+84} \lor \neg \left(x \leq 3.9 \cdot 10^{-104}\right):\\ \;\;\;\;z + 3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]

Alternative 7: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \color{blue}{\left(\left(\left(x + y\right) + y\right) + x\right) + \left(z + x\right)} \]

   2. +-commutative 99.9%

      \[\leadsto \left(\left(\color{blue}{\left(y + x\right)} + y\right) + x\right) + \left(z + x\right) \]

   3. associate-+l+ 99.9%

      \[\leadsto \color{blue}{\left(\left(y + x\right) + \left(y + x\right)\right)} + \left(z + x\right) \]

   4. count-2 99.9%

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

   5. +-commutative 99.9%

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

   6. +-commutative 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto 2 \cdot \left(x + y\right) + \left(z + x\right) \]

Alternative 8: 53.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+94} \lor \neg \left(y \leq 1.26 \cdot 10^{+25}\right):\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.4e+94) (not (<= y 1.26e+25))) (* y 2.0) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.4e+94) || !(y <= 1.26e+25)) {
		tmp = y * 2.0;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.4d+94)) .or. (.not. (y <= 1.26d+25))) then
        tmp = y * 2.0d0
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.4e+94) || !(y <= 1.26e+25)) {
		tmp = y * 2.0;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.4e+94) or not (y <= 1.26e+25):
		tmp = y * 2.0
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.4e+94) || !(y <= 1.26e+25))
		tmp = Float64(y * 2.0);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.4e+94) || ~((y <= 1.26e+25)))
		tmp = y * 2.0;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.4e+94], N[Not[LessEqual[y, 1.26e+25]], $MachinePrecision]], N[(y * 2.0), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999983e94 or 1.26000000000000008e25 < y

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.9%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.9%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.9%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.9%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.9%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.9%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.9%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.9%

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

      10. distribute-lft1-in 99.9%

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

      11. fma-def 100.0%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 100.0%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in y around inf 55.9%

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

   if -2.39999999999999983e94 < y < 1.26000000000000008e25

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

      2. associate-+l+ 99.8%

         \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

      3. +-commutative 99.8%

         \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

      4. +-commutative 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

      5. associate-+l+ 99.8%

         \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

      6. associate-+r+ 99.8%

         \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

      7. associate-+r+ 99.8%

         \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

      8. +-commutative 99.8%

         \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

      9. count-2 99.8%

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

      10. distribute-lft1-in 99.8%

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

      11. fma-def 99.9%

          \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

      12. metadata-eval 99.9%

          \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

      13. count-2 99.9%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

      14. *-commutative 99.9%

          \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

   4. Taylor expanded in z around inf 49.0%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+94} \lor \neg \left(y \leq 1.26 \cdot 10^{+25}\right):\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 9: 35.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 z)

C

double code(double x, double y, double z) {
	return z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z
end function

Java

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

Python

def code(x, y, z):
	return z

Julia

function code(x, y, z)
	return z
end

MATLAB

function tmp = code(x, y, z)
	tmp = z;
end

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
2. Step-by-step derivation

   1. +-commutative 99.9%

      \[\leadsto \color{blue}{\left(z + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} + x \]

   2. associate-+l+ 99.9%

      \[\leadsto \color{blue}{z + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + x\right)} \]

   3. +-commutative 99.9%

      \[\leadsto z + \color{blue}{\left(x + \left(\left(\left(x + y\right) + y\right) + x\right)\right)} \]

   4. +-commutative 99.9%

      \[\leadsto z + \left(x + \color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)}\right) \]

   5. associate-+l+ 99.9%

      \[\leadsto z + \left(x + \left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right)\right) \]

   6. associate-+r+ 99.9%

      \[\leadsto z + \left(x + \color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)}\right) \]

   7. associate-+r+ 99.9%

      \[\leadsto z + \color{blue}{\left(\left(x + \left(x + x\right)\right) + \left(y + y\right)\right)} \]

   8. +-commutative 99.9%

      \[\leadsto z + \left(\color{blue}{\left(\left(x + x\right) + x\right)} + \left(y + y\right)\right) \]

   9. count-2 99.9%

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

   10. distribute-lft1-in 99.9%

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

   11. fma-def 100.0%

       \[\leadsto z + \color{blue}{\mathsf{fma}\left(2 + 1, x, y + y\right)} \]

   12. metadata-eval 100.0%

       \[\leadsto z + \mathsf{fma}\left(\color{blue}{3}, x, y + y\right) \]

   13. count-2 100.0%

       \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{2 \cdot y}\right) \]

   14. *-commutative 100.0%

       \[\leadsto z + \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{z + \mathsf{fma}\left(3, x, y \cdot 2\right)} \]

4. Taylor expanded in z around inf 37.5%

   \[\leadsto \color{blue}{z} \]

5. Final simplification 37.5%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  :precision binary64
  (+ (+ (+ (+ (+ x y) y) x) z) x))