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

Percentage Accurate: 99.9% → 100.0%

Time: 4.6s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative 99.9%

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

   3. associate-+l+ 99.9%

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

   4. associate-+r+ 99.9%

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

   5. count-2 99.9%

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

   6. associate-+l+ 99.9%

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

   7. associate-+r+ 99.9%

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

   8. distribute-rgt1-in 99.9%

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

   9. *-commutative 99.9%

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

   10. fma-def 100.0%

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

   11. metadata-eval 100.0%

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

   12. count-2 100.0%

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

   13. *-commutative 100.0%

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

   14. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(z + N[(x * 3.0 + 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%

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

   2. +-commutative 99.9%

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

   3. associate-+l+ 99.9%

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

   4. associate-+r+ 99.9%

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

   5. count-2 99.9%

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

   6. associate-+l+ 99.9%

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

   7. associate-+r+ 99.9%

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

   8. distribute-rgt1-in 99.9%

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

   9. *-commutative 99.9%

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

   10. fma-def 100.0%

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

   11. metadata-eval 100.0%

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

   12. count-2 100.0%

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

   13. *-commutative 100.0%

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

   14. fma-def 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.9%

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

   2. fma-udef 99.9%

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

   3. associate-+r+ 99.9%

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

   4. fma-def 100.0%

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

   5. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 52.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.06 \cdot 10^{+36}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-217}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-233}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-57}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+110}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.06e+36)
   z
   (if (<= z -1.1e-217)
     (* y 2.0)
     (if (<= z 1.35e-233)
       (* x 3.0)
       (if (<= z 6.2e-188)
         (* y 2.0)
         (if (<= z 2.8e-57) (* x 3.0) (if (<= z 3.4e+110) (* y 2.0) z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.06e+36) {
		tmp = z;
	} else if (z <= -1.1e-217) {
		tmp = y * 2.0;
	} else if (z <= 1.35e-233) {
		tmp = x * 3.0;
	} else if (z <= 6.2e-188) {
		tmp = y * 2.0;
	} else if (z <= 2.8e-57) {
		tmp = x * 3.0;
	} else if (z <= 3.4e+110) {
		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 (z <= (-2.06d+36)) then
        tmp = z
    else if (z <= (-1.1d-217)) then
        tmp = y * 2.0d0
    else if (z <= 1.35d-233) then
        tmp = x * 3.0d0
    else if (z <= 6.2d-188) then
        tmp = y * 2.0d0
    else if (z <= 2.8d-57) then
        tmp = x * 3.0d0
    else if (z <= 3.4d+110) 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 (z <= -2.06e+36) {
		tmp = z;
	} else if (z <= -1.1e-217) {
		tmp = y * 2.0;
	} else if (z <= 1.35e-233) {
		tmp = x * 3.0;
	} else if (z <= 6.2e-188) {
		tmp = y * 2.0;
	} else if (z <= 2.8e-57) {
		tmp = x * 3.0;
	} else if (z <= 3.4e+110) {
		tmp = y * 2.0;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.06e+36:
		tmp = z
	elif z <= -1.1e-217:
		tmp = y * 2.0
	elif z <= 1.35e-233:
		tmp = x * 3.0
	elif z <= 6.2e-188:
		tmp = y * 2.0
	elif z <= 2.8e-57:
		tmp = x * 3.0
	elif z <= 3.4e+110:
		tmp = y * 2.0
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.06e+36)
		tmp = z;
	elseif (z <= -1.1e-217)
		tmp = Float64(y * 2.0);
	elseif (z <= 1.35e-233)
		tmp = Float64(x * 3.0);
	elseif (z <= 6.2e-188)
		tmp = Float64(y * 2.0);
	elseif (z <= 2.8e-57)
		tmp = Float64(x * 3.0);
	elseif (z <= 3.4e+110)
		tmp = Float64(y * 2.0);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.06e+36)
		tmp = z;
	elseif (z <= -1.1e-217)
		tmp = y * 2.0;
	elseif (z <= 1.35e-233)
		tmp = x * 3.0;
	elseif (z <= 6.2e-188)
		tmp = y * 2.0;
	elseif (z <= 2.8e-57)
		tmp = x * 3.0;
	elseif (z <= 3.4e+110)
		tmp = y * 2.0;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.06e+36], z, If[LessEqual[z, -1.1e-217], N[(y * 2.0), $MachinePrecision], If[LessEqual[z, 1.35e-233], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, 6.2e-188], N[(y * 2.0), $MachinePrecision], If[LessEqual[z, 2.8e-57], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, 3.4e+110], N[(y * 2.0), $MachinePrecision], z]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.06000000000000006e36 or 3.4000000000000001e110 < z

   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%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 77.0%

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

   if -2.06000000000000006e36 < z < -1.09999999999999991e-217 or 1.35e-233 < z < 6.2000000000000004e-188 or 2.7999999999999999e-57 < z < 3.4000000000000001e110

   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%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 50.2%

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

   if -1.09999999999999991e-217 < z < 1.35e-233 or 6.2000000000000004e-188 < z < 2.7999999999999999e-57

   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%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. count-2 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 74.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.06 \cdot 10^{+36}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-217}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-233}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-57}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+110}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{+125}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+49} \lor \neg \left(x \leq -4.7 \cdot 10^{+36}\right) \land x \leq 2.9 \cdot 10^{+154}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.05e+125)
   (* x 3.0)
   (if (or (<= x -1.05e+49) (and (not (<= x -4.7e+36)) (<= x 2.9e+154)))
     (+ z (* y 2.0))
     (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.05e+125) {
		tmp = x * 3.0;
	} else if ((x <= -1.05e+49) || (!(x <= -4.7e+36) && (x <= 2.9e+154))) {
		tmp = z + (y * 2.0);
	} else {
		tmp = x * 3.0;
	}
	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 <= (-3.05d+125)) then
        tmp = x * 3.0d0
    else if ((x <= (-1.05d+49)) .or. (.not. (x <= (-4.7d+36))) .and. (x <= 2.9d+154)) then
        tmp = z + (y * 2.0d0)
    else
        tmp = x * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.05e+125) {
		tmp = x * 3.0;
	} else if ((x <= -1.05e+49) || (!(x <= -4.7e+36) && (x <= 2.9e+154))) {
		tmp = z + (y * 2.0);
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.05e+125:
		tmp = x * 3.0
	elif (x <= -1.05e+49) or (not (x <= -4.7e+36) and (x <= 2.9e+154)):
		tmp = z + (y * 2.0)
	else:
		tmp = x * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.05e+125)
		tmp = Float64(x * 3.0);
	elseif ((x <= -1.05e+49) || (!(x <= -4.7e+36) && (x <= 2.9e+154)))
		tmp = Float64(z + Float64(y * 2.0));
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.05e+125)
		tmp = x * 3.0;
	elseif ((x <= -1.05e+49) || (~((x <= -4.7e+36)) && (x <= 2.9e+154)))
		tmp = z + (y * 2.0);
	else
		tmp = x * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.05e+125], N[(x * 3.0), $MachinePrecision], If[Or[LessEqual[x, -1.05e+49], And[N[Not[LessEqual[x, -4.7e+36]], $MachinePrecision], LessEqual[x, 2.9e+154]]], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.04999999999999988e125 or -1.05000000000000005e49 < x < -4.69999999999999989e36 or 2.89999999999999979e154 < x

   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%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 85.2%

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

   if -3.04999999999999988e125 < x < -1.05000000000000005e49 or -4.69999999999999989e36 < x < 2.89999999999999979e154

   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%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 84.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{+125}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+49} \lor \neg \left(x \leq -4.7 \cdot 10^{+36}\right) \land x \leq 2.9 \cdot 10^{+154}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]

Alternative 5: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+62} \lor \neg \left(y \leq 1.56 \cdot 10^{+37}\right):\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(x + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.2e+62) (not (<= y 1.56e+37)))
   (+ z (* y 2.0))
   (+ (* x 2.0) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.2e+62) || !(y <= 1.56e+37)) {
		tmp = z + (y * 2.0);
	} else {
		tmp = (x * 2.0) + (x + 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 <= (-3.2d+62)) .or. (.not. (y <= 1.56d+37))) then
        tmp = z + (y * 2.0d0)
    else
        tmp = (x * 2.0d0) + (x + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.2e+62) || !(y <= 1.56e+37)) {
		tmp = z + (y * 2.0);
	} else {
		tmp = (x * 2.0) + (x + z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.2e+62) or not (y <= 1.56e+37):
		tmp = z + (y * 2.0)
	else:
		tmp = (x * 2.0) + (x + z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.2e+62) || !(y <= 1.56e+37))
		tmp = Float64(z + Float64(y * 2.0));
	else
		tmp = Float64(Float64(x * 2.0) + Float64(x + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.2e+62) || ~((y <= 1.56e+37)))
		tmp = z + (y * 2.0);
	else
		tmp = (x * 2.0) + (x + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.2e+62], N[Not[LessEqual[y, 1.56e+37]], $MachinePrecision]], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] + N[(x + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999984e62 or 1.56000000000000008e37 < 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%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 86.9%

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

   if -3.19999999999999984e62 < y < 1.56000000000000008e37

   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%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 91.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+62} \lor \neg \left(y \leq 1.56 \cdot 10^{+37}\right):\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(x + z\right)\\ \end{array} \]

Alternative 6: 85.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+58} \lor \neg \left(y \leq 7.8 \cdot 10^{+38}\right):\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8.2e+58) (not (<= y 7.8e+38)))
   (+ z (* y 2.0))
   (+ z (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.2e+58) || !(y <= 7.8e+38)) {
		tmp = z + (y * 2.0);
	} else {
		tmp = z + (x * 3.0);
	}
	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 ((y <= (-8.2d+58)) .or. (.not. (y <= 7.8d+38))) then
        tmp = z + (y * 2.0d0)
    else
        tmp = z + (x * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.2e+58) || !(y <= 7.8e+38)) {
		tmp = z + (y * 2.0);
	} else {
		tmp = z + (x * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8.2e+58) or not (y <= 7.8e+38):
		tmp = z + (y * 2.0)
	else:
		tmp = z + (x * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8.2e+58) || !(y <= 7.8e+38))
		tmp = Float64(z + Float64(y * 2.0));
	else
		tmp = Float64(z + Float64(x * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8.2e+58) || ~((y <= 7.8e+38)))
		tmp = z + (y * 2.0);
	else
		tmp = z + (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8.2e+58], N[Not[LessEqual[y, 7.8e+38]], $MachinePrecision]], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.2e58 or 7.80000000000000047e38 < 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%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 86.9%

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

   if -8.2e58 < y < 7.80000000000000047e38

   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%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r+ 99.9%

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

      5. count-2 99.9%

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

      6. associate-+l+ 99.9%

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

      7. associate-+r+ 99.9%

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

      8. distribute-rgt1-in 99.9%

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

      9. *-commutative 99.9%

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

      10. fma-def 100.0%

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

      11. metadata-eval 100.0%

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

      12. count-2 100.0%

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

      13. *-commutative 100.0%

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

      14. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 91.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+58} \lor \neg \left(y \leq 7.8 \cdot 10^{+38}\right):\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \]

Alternative 7: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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 + (z + (2.0d0 * (x + y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-+l+ 99.9%

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

   3. +-commutative 99.9%

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

   4. count-2 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 8: 52.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+116}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+39}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e+116) (* y 2.0) (if (<= y 1.45e+39) z (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+116) {
		tmp = y * 2.0;
	} else if (y <= 1.45e+39) {
		tmp = z;
	} else {
		tmp = 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 (y <= (-1d+116)) then
        tmp = y * 2.0d0
    else if (y <= 1.45d+39) then
        tmp = z
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+116) {
		tmp = y * 2.0;
	} else if (y <= 1.45e+39) {
		tmp = z;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e+116:
		tmp = y * 2.0
	elif y <= 1.45e+39:
		tmp = z
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e+116)
		tmp = Float64(y * 2.0);
	elseif (y <= 1.45e+39)
		tmp = z;
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e+116)
		tmp = y * 2.0;
	elseif (y <= 1.45e+39)
		tmp = z;
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e+116], N[(y * 2.0), $MachinePrecision], If[LessEqual[y, 1.45e+39], z, N[(y * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000002e116 or 1.45000000000000015e39 < 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%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 71.6%

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

   if -1.00000000000000002e116 < y < 1.45000000000000015e39

   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%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 46.4%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+116}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+39}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 9: 34.0% 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%

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

   2. associate-+l+ 99.9%

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

   3. +-commutative 99.9%

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

   4. count-2 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in z around inf 36.3%

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

5. Final simplification 36.3%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023257 
(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))