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

Percentage Accurate: 99.9% → 100.0%

Time: 8.5s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(z + N[(x * 3.0 + N[(2.0 * y), $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. remove-double-neg 99.9%

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

   4. distribute-neg-in 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. remove-double-neg 99.9%

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

   7. sub-neg 99.9%

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

   8. associate-+l+ 99.9%

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

   9. +-commutative 99.9%

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

   10. associate-+r+ 99.9%

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

   11. associate--l+ 99.9%

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

   12. count-2 99.9%

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

   13. *-commutative 99.9%

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

   14. fma-def 99.9%

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

   15. count-2 99.9%

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

   16. neg-mul-1 99.9%

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

   17. distribute-rgt-out-- 99.9%

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

   18. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. add-log-exp 22.0%

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

   2. *-un-lft-identity 22.0%

      \[\leadsto z + \log \color{blue}{\left(1 \cdot e^{\mathsf{fma}\left(y, 2, x \cdot 3\right)}\right)} \]

   3. log-prod 22.0%

      \[\leadsto z + \color{blue}{\left(\log 1 + \log \left(e^{\mathsf{fma}\left(y, 2, x \cdot 3\right)}\right)\right)} \]

   4. metadata-eval 22.0%

      \[\leadsto z + \left(\color{blue}{0} + \log \left(e^{\mathsf{fma}\left(y, 2, x \cdot 3\right)}\right)\right) \]

   5. add-log-exp 99.9%

      \[\leadsto z + \left(0 + \color{blue}{\mathsf{fma}\left(y, 2, x \cdot 3\right)}\right) \]

   6. fma-udef 99.9%

      \[\leadsto z + \left(0 + \color{blue}{\left(y \cdot 2 + x \cdot 3\right)}\right) \]

   7. +-commutative 99.9%

      \[\leadsto z + \left(0 + \color{blue}{\left(x \cdot 3 + y \cdot 2\right)}\right) \]

   8. fma-def 100.0%

      \[\leadsto z + \left(0 + \color{blue}{\mathsf{fma}\left(x, 3, y \cdot 2\right)}\right) \]

   9. *-commutative 100.0%

      \[\leadsto z + \left(0 + \mathsf{fma}\left(x, 3, \color{blue}{2 \cdot y}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto z + \color{blue}{\left(0 + \mathsf{fma}\left(x, 3, 2 \cdot y\right)\right)} \]

6. Final simplification 100.0%

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

Alternative 2: 53.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+20}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-296}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-84}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-68}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+55}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5e+20)
   z
   (if (<= z -8.2e-296)
     (* x 3.0)
     (if (<= z 2.3e-84)
       (* 2.0 y)
       (if (<= z 6e-68) (* x 3.0) (if (<= z 1.18e+55) (* 2.0 y) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+20) {
		tmp = z;
	} else if (z <= -8.2e-296) {
		tmp = x * 3.0;
	} else if (z <= 2.3e-84) {
		tmp = 2.0 * y;
	} else if (z <= 6e-68) {
		tmp = x * 3.0;
	} else if (z <= 1.18e+55) {
		tmp = 2.0 * y;
	} 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 <= (-5d+20)) then
        tmp = z
    else if (z <= (-8.2d-296)) then
        tmp = x * 3.0d0
    else if (z <= 2.3d-84) then
        tmp = 2.0d0 * y
    else if (z <= 6d-68) then
        tmp = x * 3.0d0
    else if (z <= 1.18d+55) then
        tmp = 2.0d0 * y
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+20) {
		tmp = z;
	} else if (z <= -8.2e-296) {
		tmp = x * 3.0;
	} else if (z <= 2.3e-84) {
		tmp = 2.0 * y;
	} else if (z <= 6e-68) {
		tmp = x * 3.0;
	} else if (z <= 1.18e+55) {
		tmp = 2.0 * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5e+20:
		tmp = z
	elif z <= -8.2e-296:
		tmp = x * 3.0
	elif z <= 2.3e-84:
		tmp = 2.0 * y
	elif z <= 6e-68:
		tmp = x * 3.0
	elif z <= 1.18e+55:
		tmp = 2.0 * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e+20)
		tmp = z;
	elseif (z <= -8.2e-296)
		tmp = Float64(x * 3.0);
	elseif (z <= 2.3e-84)
		tmp = Float64(2.0 * y);
	elseif (z <= 6e-68)
		tmp = Float64(x * 3.0);
	elseif (z <= 1.18e+55)
		tmp = Float64(2.0 * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e+20)
		tmp = z;
	elseif (z <= -8.2e-296)
		tmp = x * 3.0;
	elseif (z <= 2.3e-84)
		tmp = 2.0 * y;
	elseif (z <= 6e-68)
		tmp = x * 3.0;
	elseif (z <= 1.18e+55)
		tmp = 2.0 * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5e+20], z, If[LessEqual[z, -8.2e-296], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, 2.3e-84], N[(2.0 * y), $MachinePrecision], If[LessEqual[z, 6e-68], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, 1.18e+55], N[(2.0 * y), $MachinePrecision], z]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5e20 or 1.1799999999999999e55 < 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%

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

      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 69.2%

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

   if -5e20 < z < -8.19999999999999988e-296 or 2.29999999999999981e-84 < z < 6e-68

   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 60.2%

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

   if -8.19999999999999988e-296 < z < 2.29999999999999981e-84 or 6e-68 < z < 1.1799999999999999e55

   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 62.5%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+20}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-296}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-84}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-68}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+55}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z + 2 \cdot y\\ \mathbf{if}\;y \leq -2 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+31}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+171} \lor \neg \left(y \leq 9.8 \cdot 10^{+203}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ z (* 2.0 y))))
   (if (<= y -2e+62)
     t_0
     (if (<= y 4.5e+31)
       (+ z (* x 3.0))
       (if (or (<= y 1.1e+171) (not (<= y 9.8e+203)))
         t_0
         (/ x 0.3333333333333333))))))

C

double code(double x, double y, double z) {
	double t_0 = z + (2.0 * y);
	double tmp;
	if (y <= -2e+62) {
		tmp = t_0;
	} else if (y <= 4.5e+31) {
		tmp = z + (x * 3.0);
	} else if ((y <= 1.1e+171) || !(y <= 9.8e+203)) {
		tmp = t_0;
	} else {
		tmp = x / 0.3333333333333333;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = z + (2.0d0 * y)
    if (y <= (-2d+62)) then
        tmp = t_0
    else if (y <= 4.5d+31) then
        tmp = z + (x * 3.0d0)
    else if ((y <= 1.1d+171) .or. (.not. (y <= 9.8d+203))) then
        tmp = t_0
    else
        tmp = x / 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z + (2.0 * y);
	double tmp;
	if (y <= -2e+62) {
		tmp = t_0;
	} else if (y <= 4.5e+31) {
		tmp = z + (x * 3.0);
	} else if ((y <= 1.1e+171) || !(y <= 9.8e+203)) {
		tmp = t_0;
	} else {
		tmp = x / 0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z + (2.0 * y)
	tmp = 0
	if y <= -2e+62:
		tmp = t_0
	elif y <= 4.5e+31:
		tmp = z + (x * 3.0)
	elif (y <= 1.1e+171) or not (y <= 9.8e+203):
		tmp = t_0
	else:
		tmp = x / 0.3333333333333333
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z + Float64(2.0 * y))
	tmp = 0.0
	if (y <= -2e+62)
		tmp = t_0;
	elseif (y <= 4.5e+31)
		tmp = Float64(z + Float64(x * 3.0));
	elseif ((y <= 1.1e+171) || !(y <= 9.8e+203))
		tmp = t_0;
	else
		tmp = Float64(x / 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z + (2.0 * y);
	tmp = 0.0;
	if (y <= -2e+62)
		tmp = t_0;
	elseif (y <= 4.5e+31)
		tmp = z + (x * 3.0);
	elseif ((y <= 1.1e+171) || ~((y <= 9.8e+203)))
		tmp = t_0;
	else
		tmp = x / 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e+62], t$95$0, If[LessEqual[y, 4.5e+31], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.1e+171], N[Not[LessEqual[y, 9.8e+203]], $MachinePrecision]], t$95$0, N[(x / 0.3333333333333333), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.00000000000000007e62 or 4.4999999999999996e31 < y < 1.1e171 or 9.7999999999999995e203 < 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+ 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 x around 0 86.6%

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

   if -2.00000000000000007e62 < y < 4.4999999999999996e31

   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. remove-double-neg 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. associate--l+ 99.9%

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

      12. count-2 99.9%

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

      13. *-commutative 99.9%

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

      14. fma-def 99.9%

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

      15. count-2 99.9%

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

      16. neg-mul-1 99.9%

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

      17. distribute-rgt-out-- 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 91.1%

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

   if 1.1e171 < y < 9.7999999999999995e203

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-+l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. associate--l+ 99.7%

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

      12. count-2 99.7%

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

      13. *-commutative 99.7%

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

      14. fma-def 99.7%

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

      15. count-2 99.7%

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

      16. neg-mul-1 99.7%

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

      17. distribute-rgt-out-- 99.7%

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

      18. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 99.7%

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

      1. flip-+ 5.8%

         \[\leadsto \color{blue}{\frac{\left(3 \cdot x\right) \cdot \left(3 \cdot x\right) - z \cdot z}{3 \cdot x - z}} \]

      2. div-sub 5.8%

         \[\leadsto \color{blue}{\frac{\left(3 \cdot x\right) \cdot \left(3 \cdot x\right)}{3 \cdot x - z} - \frac{z \cdot z}{3 \cdot x - z}} \]

      3. swap-sqr 5.8%

         \[\leadsto \frac{\color{blue}{\left(3 \cdot 3\right) \cdot \left(x \cdot x\right)}}{3 \cdot x - z} - \frac{z \cdot z}{3 \cdot x - z} \]

      4. metadata-eval 5.8%

         \[\leadsto \frac{\color{blue}{9} \cdot \left(x \cdot x\right)}{3 \cdot x - z} - \frac{z \cdot z}{3 \cdot x - z} \]

   6. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot x\right)}{3 \cdot x - z} - \frac{z \cdot z}{3 \cdot x - z}} \]
   7. Step-by-step derivation

      1. div-sub 5.8%

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot x\right) - z \cdot z}{3 \cdot x - z}} \]

      2. *-commutative 5.8%

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot 9} - z \cdot z}{3 \cdot x - z} \]

      3. associate-*l* 5.8%

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot 9\right)} - z \cdot z}{3 \cdot x - z} \]

   8. Simplified 5.8%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot 9\right) - z \cdot z}{3 \cdot x - z}} \]

   9. Taylor expanded in x around inf 99.7%

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

       1. *-commutative 99.7%

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

       2. /-rgt-identity 99.7%

          \[\leadsto \color{blue}{\frac{x \cdot 3}{1}} \]

       3. associate-/l* 100.0%

          \[\leadsto \color{blue}{\frac{x}{\frac{1}{3}}} \]

       4. metadata-eval 100.0%

          \[\leadsto \frac{x}{\color{blue}{0.3333333333333333}} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{x}{0.3333333333333333}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+62}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+31}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+171} \lor \neg \left(y \leq 9.8 \cdot 10^{+203}\right):\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.3333333333333333}\\ \end{array} \]

Alternative 4: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+63}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.2e+63)
   (+ z (* 2.0 y))
   (if (<= y 1.6e+21) (+ z (* x 3.0)) (+ x (* 2.0 (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2e+63) {
		tmp = z + (2.0 * y);
	} else if (y <= 1.6e+21) {
		tmp = z + (x * 3.0);
	} 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 <= (-1.2d+63)) then
        tmp = z + (2.0d0 * y)
    else if (y <= 1.6d+21) then
        tmp = z + (x * 3.0d0)
    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 <= -1.2e+63) {
		tmp = z + (2.0 * y);
	} else if (y <= 1.6e+21) {
		tmp = z + (x * 3.0);
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.2e+63:
		tmp = z + (2.0 * y)
	elif y <= 1.6e+21:
		tmp = z + (x * 3.0)
	else:
		tmp = x + (2.0 * (x + y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.2e+63)
		tmp = Float64(z + Float64(2.0 * y));
	elseif (y <= 1.6e+21)
		tmp = Float64(z + Float64(x * 3.0));
	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 <= -1.2e+63)
		tmp = z + (2.0 * y);
	elseif (y <= 1.6e+21)
		tmp = z + (x * 3.0);
	else
		tmp = x + (2.0 * (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.2e+63], N[(z + N[(2.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+21], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e63

   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+ 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 x around 0 85.2%

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

   if -1.2e63 < y < 1.6e21

   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+ 99.9%

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

      3. remove-double-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-+l+ 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. associate--l+ 99.9%

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

      12. count-2 99.9%

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

      13. *-commutative 99.9%

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

      14. fma-def 99.9%

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

      15. count-2 99.9%

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

      16. neg-mul-1 99.9%

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

      17. distribute-rgt-out-- 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 91.0%

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

   if 1.6e21 < 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 z around 0 92.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+63}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 5: 80.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+153}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e+153)
   (* x 3.0)
   (if (<= x 3.2e+106) (+ z (* 2.0 y)) (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+153) {
		tmp = x * 3.0;
	} else if (x <= 3.2e+106) {
		tmp = z + (2.0 * y);
	} 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 <= (-1.8d+153)) then
        tmp = x * 3.0d0
    else if (x <= 3.2d+106) then
        tmp = z + (2.0d0 * y)
    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 <= -1.8e+153) {
		tmp = x * 3.0;
	} else if (x <= 3.2e+106) {
		tmp = z + (2.0 * y);
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.8e+153:
		tmp = x * 3.0
	elif x <= 3.2e+106:
		tmp = z + (2.0 * y)
	else:
		tmp = x * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.8e+153)
		tmp = Float64(x * 3.0);
	elseif (x <= 3.2e+106)
		tmp = Float64(z + Float64(2.0 * y));
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.8e+153)
		tmp = x * 3.0;
	elseif (x <= 3.2e+106)
		tmp = z + (2.0 * y);
	else
		tmp = x * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.8e+153], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, 3.2e+106], N[(z + N[(2.0 * y), $MachinePrecision]), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e153 or 3.1999999999999998e106 < 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%

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

      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 74.4%

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

   if -1.8e153 < x < 3.1999999999999998e106

   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 81.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+153}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]

Alternative 6: 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 7: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * 3.0) + (z + (2.0 * 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 * 3.0d0) + (z + (2.0d0 * y))
end function

Java

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

Python

def code(x, y, z):
	return (x * 3.0) + (z + (2.0 * y))

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 3.0) + (z + (2.0 * y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * 3.0), $MachinePrecision] + N[(z + N[(2.0 * y), $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. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

   \[\leadsto x \cdot 3 + \left(z + 2 \cdot y\right) \]

Alternative 8: 52.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+21}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.7e+36) (* 2.0 y) (if (<= y 2.2e+21) z (* 2.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.7e+36) {
		tmp = 2.0 * y;
	} else if (y <= 2.2e+21) {
		tmp = z;
	} else {
		tmp = 2.0 * 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 <= (-3.7d+36)) then
        tmp = 2.0d0 * y
    else if (y <= 2.2d+21) then
        tmp = z
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.7e+36) {
		tmp = 2.0 * y;
	} else if (y <= 2.2e+21) {
		tmp = z;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.7e+36:
		tmp = 2.0 * y
	elif y <= 2.2e+21:
		tmp = z
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.7e+36)
		tmp = Float64(2.0 * y);
	elseif (y <= 2.2e+21)
		tmp = z;
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.7e+36)
		tmp = 2.0 * y;
	elseif (y <= 2.2e+21)
		tmp = z;
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.7e+36], N[(2.0 * y), $MachinePrecision], If[LessEqual[y, 2.2e+21], z, N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.70000000000000029e36 or 2.2e21 < 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 64.4%

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

   if -3.70000000000000029e36 < y < 2.2e21

   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 z around inf 51.5%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+21}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]

Alternative 9: 34.7% 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 35.9%

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

5. Final simplification 35.9%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023279 
(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))