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

Percentage Accurate: 99.9% → 99.9%

Time: 5.7s

Alternatives: 10

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: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(z + N[(x * 3.0 + N[(y + 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. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. *-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

   1. fma-def 100.0%

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

   2. add-log-exp 53.6%

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

   3. *-commutative 53.6%

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

   4. exp-lft-sqr 53.5%

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

   5. log-prod 53.9%

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

   6. add-log-exp 54.6%

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

   7. add-log-exp 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 52.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+38}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{-30}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-38}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-122}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-174}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-232}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-301}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{+75}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.55e+38)
   z
   (if (<= z -2.85e-30)
     (* y 2.0)
     (if (<= z -2.9e-38)
       z
       (if (<= z -8.5e-122)
         (* x 3.0)
         (if (<= z -3.7e-174)
           (* y 2.0)
           (if (<= z -4.7e-232)
             (* x 3.0)
             (if (<= z 2.25e-301)
               (* y 2.0)
               (if (<= z 1.28e+75) (* x 3.0) z)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.55e+38) {
		tmp = z;
	} else if (z <= -2.85e-30) {
		tmp = y * 2.0;
	} else if (z <= -2.9e-38) {
		tmp = z;
	} else if (z <= -8.5e-122) {
		tmp = x * 3.0;
	} else if (z <= -3.7e-174) {
		tmp = y * 2.0;
	} else if (z <= -4.7e-232) {
		tmp = x * 3.0;
	} else if (z <= 2.25e-301) {
		tmp = y * 2.0;
	} else if (z <= 1.28e+75) {
		tmp = x * 3.0;
	} 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 <= (-1.55d+38)) then
        tmp = z
    else if (z <= (-2.85d-30)) then
        tmp = y * 2.0d0
    else if (z <= (-2.9d-38)) then
        tmp = z
    else if (z <= (-8.5d-122)) then
        tmp = x * 3.0d0
    else if (z <= (-3.7d-174)) then
        tmp = y * 2.0d0
    else if (z <= (-4.7d-232)) then
        tmp = x * 3.0d0
    else if (z <= 2.25d-301) then
        tmp = y * 2.0d0
    else if (z <= 1.28d+75) then
        tmp = x * 3.0d0
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.55e+38) {
		tmp = z;
	} else if (z <= -2.85e-30) {
		tmp = y * 2.0;
	} else if (z <= -2.9e-38) {
		tmp = z;
	} else if (z <= -8.5e-122) {
		tmp = x * 3.0;
	} else if (z <= -3.7e-174) {
		tmp = y * 2.0;
	} else if (z <= -4.7e-232) {
		tmp = x * 3.0;
	} else if (z <= 2.25e-301) {
		tmp = y * 2.0;
	} else if (z <= 1.28e+75) {
		tmp = x * 3.0;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.55e+38:
		tmp = z
	elif z <= -2.85e-30:
		tmp = y * 2.0
	elif z <= -2.9e-38:
		tmp = z
	elif z <= -8.5e-122:
		tmp = x * 3.0
	elif z <= -3.7e-174:
		tmp = y * 2.0
	elif z <= -4.7e-232:
		tmp = x * 3.0
	elif z <= 2.25e-301:
		tmp = y * 2.0
	elif z <= 1.28e+75:
		tmp = x * 3.0
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.55e+38)
		tmp = z;
	elseif (z <= -2.85e-30)
		tmp = Float64(y * 2.0);
	elseif (z <= -2.9e-38)
		tmp = z;
	elseif (z <= -8.5e-122)
		tmp = Float64(x * 3.0);
	elseif (z <= -3.7e-174)
		tmp = Float64(y * 2.0);
	elseif (z <= -4.7e-232)
		tmp = Float64(x * 3.0);
	elseif (z <= 2.25e-301)
		tmp = Float64(y * 2.0);
	elseif (z <= 1.28e+75)
		tmp = Float64(x * 3.0);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.55e+38)
		tmp = z;
	elseif (z <= -2.85e-30)
		tmp = y * 2.0;
	elseif (z <= -2.9e-38)
		tmp = z;
	elseif (z <= -8.5e-122)
		tmp = x * 3.0;
	elseif (z <= -3.7e-174)
		tmp = y * 2.0;
	elseif (z <= -4.7e-232)
		tmp = x * 3.0;
	elseif (z <= 2.25e-301)
		tmp = y * 2.0;
	elseif (z <= 1.28e+75)
		tmp = x * 3.0;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.55e+38], z, If[LessEqual[z, -2.85e-30], N[(y * 2.0), $MachinePrecision], If[LessEqual[z, -2.9e-38], z, If[LessEqual[z, -8.5e-122], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, -3.7e-174], N[(y * 2.0), $MachinePrecision], If[LessEqual[z, -4.7e-232], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, 2.25e-301], N[(y * 2.0), $MachinePrecision], If[LessEqual[z, 1.28e+75], N[(x * 3.0), $MachinePrecision], z]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55000000000000009e38 or -2.84999999999999989e-30 < z < -2.89999999999999994e-38 or 1.27999999999999998e75 < 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 72.5%

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

   if -1.55000000000000009e38 < z < -2.84999999999999989e-30 or -8.50000000000000003e-122 < z < -3.7000000000000001e-174 or -4.70000000000000035e-232 < z < 2.2500000000000001e-301

   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 72.3%

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

   if -2.89999999999999994e-38 < z < -8.50000000000000003e-122 or -3.7000000000000001e-174 < z < -4.70000000000000035e-232 or 2.2500000000000001e-301 < z < 1.27999999999999998e75

   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 58.8%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+38}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{-30}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-38}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-122}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-174}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-232}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-301}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{+75}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z + x \cdot 3\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-65}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-27}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ z (* x 3.0))))
   (if (<= x -1.7e+99)
     t_0
     (if (<= x -2.65e-65)
       (+ x (* 2.0 (+ x y)))
       (if (<= x 3e-27) (+ z (* y 2.0)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z + (x * 3.0);
	double tmp;
	if (x <= -1.7e+99) {
		tmp = t_0;
	} else if (x <= -2.65e-65) {
		tmp = x + (2.0 * (x + y));
	} else if (x <= 3e-27) {
		tmp = z + (y * 2.0);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = z + (x * 3.0d0)
    if (x <= (-1.7d+99)) then
        tmp = t_0
    else if (x <= (-2.65d-65)) then
        tmp = x + (2.0d0 * (x + y))
    else if (x <= 3d-27) then
        tmp = z + (y * 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z + (x * 3.0);
	double tmp;
	if (x <= -1.7e+99) {
		tmp = t_0;
	} else if (x <= -2.65e-65) {
		tmp = x + (2.0 * (x + y));
	} else if (x <= 3e-27) {
		tmp = z + (y * 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z + (x * 3.0)
	tmp = 0
	if x <= -1.7e+99:
		tmp = t_0
	elif x <= -2.65e-65:
		tmp = x + (2.0 * (x + y))
	elif x <= 3e-27:
		tmp = z + (y * 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z + Float64(x * 3.0))
	tmp = 0.0
	if (x <= -1.7e+99)
		tmp = t_0;
	elseif (x <= -2.65e-65)
		tmp = Float64(x + Float64(2.0 * Float64(x + y)));
	elseif (x <= 3e-27)
		tmp = Float64(z + Float64(y * 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z + (x * 3.0);
	tmp = 0.0;
	if (x <= -1.7e+99)
		tmp = t_0;
	elseif (x <= -2.65e-65)
		tmp = x + (2.0 * (x + y));
	elseif (x <= 3e-27)
		tmp = z + (y * 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e+99], t$95$0, If[LessEqual[x, -2.65e-65], N[(x + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e-27], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999992e99 or 3.0000000000000001e-27 < x

   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+ 99.8%

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

      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+ 99.7%

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

      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+ 99.8%

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

      11. associate--l+ 99.8%

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

      12. count-2 99.8%

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

      13. *-commutative 99.8%

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

      14. fma-def 99.8%

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

      15. count-2 99.8%

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

      16. neg-mul-1 99.8%

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

      17. distribute-rgt-out-- 99.8%

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

      18. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 92.7%

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

   if -1.69999999999999992e99 < x < -2.65000000000000019e-65

   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 77.5%

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

   if -2.65000000000000019e-65 < x < 3.0000000000000001e-27

   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 x around 0 95.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+99}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-65}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-27}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \]

Alternative 4: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+98}:\\ \;\;\;\;\left(z + x\right) + x \cdot 2\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-65}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-27}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.4e+98)
   (+ (+ z x) (* x 2.0))
   (if (<= x -2.65e-65)
     (+ x (* 2.0 (+ x y)))
     (if (<= x 4.2e-27) (+ z (* y 2.0)) (+ z (* x 3.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.4e+98) {
		tmp = (z + x) + (x * 2.0);
	} else if (x <= -2.65e-65) {
		tmp = x + (2.0 * (x + y));
	} else if (x <= 4.2e-27) {
		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 (x <= (-5.4d+98)) then
        tmp = (z + x) + (x * 2.0d0)
    else if (x <= (-2.65d-65)) then
        tmp = x + (2.0d0 * (x + y))
    else if (x <= 4.2d-27) 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 (x <= -5.4e+98) {
		tmp = (z + x) + (x * 2.0);
	} else if (x <= -2.65e-65) {
		tmp = x + (2.0 * (x + y));
	} else if (x <= 4.2e-27) {
		tmp = z + (y * 2.0);
	} else {
		tmp = z + (x * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.4e+98:
		tmp = (z + x) + (x * 2.0)
	elif x <= -2.65e-65:
		tmp = x + (2.0 * (x + y))
	elif x <= 4.2e-27:
		tmp = z + (y * 2.0)
	else:
		tmp = z + (x * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.4e+98)
		tmp = Float64(Float64(z + x) + Float64(x * 2.0));
	elseif (x <= -2.65e-65)
		tmp = Float64(x + Float64(2.0 * Float64(x + y)));
	elseif (x <= 4.2e-27)
		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 (x <= -5.4e+98)
		tmp = (z + x) + (x * 2.0);
	elseif (x <= -2.65e-65)
		tmp = x + (2.0 * (x + y));
	elseif (x <= 4.2e-27)
		tmp = z + (y * 2.0);
	else
		tmp = z + (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.4e+98], N[(N[(z + x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.65e-65], N[(x + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-27], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.4e98

   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%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. count-2 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 96.3%

      \[\leadsto \color{blue}{x + \left(z + 2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 96.3%

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

      2. +-commutative 96.3%

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

   6. Simplified 96.3%

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

   if -5.4e98 < x < -2.65000000000000019e-65

   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 77.5%

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

   if -2.65000000000000019e-65 < x < 4.20000000000000031e-27

   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 x around 0 95.8%

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

   if 4.20000000000000031e-27 < x

   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+ 99.8%

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

      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+ 99.8%

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

      12. count-2 99.8%

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

      13. *-commutative 99.8%

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

      14. fma-def 99.8%

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

      15. count-2 99.8%

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

      16. neg-mul-1 99.8%

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

      17. distribute-rgt-out-- 99.8%

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

      18. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 90.8%

      \[\leadsto \color{blue}{z + 3 \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+98}:\\ \;\;\;\;\left(z + x\right) + x \cdot 2\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-65}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-27}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \]

Alternative 5: 84.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.65 \cdot 10^{+37} \lor \neg \left(x \leq 5.8 \cdot 10^{-32}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.65e+37) (not (<= x 5.8e-32)))
   (+ z (* x 3.0))
   (+ z (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.65e+37) || !(x <= 5.8e-32)) {
		tmp = z + (x * 3.0);
	} 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 <= (-3.65d+37)) .or. (.not. (x <= 5.8d-32))) then
        tmp = z + (x * 3.0d0)
    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 <= -3.65e+37) || !(x <= 5.8e-32)) {
		tmp = z + (x * 3.0);
	} else {
		tmp = z + (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.65e+37) or not (x <= 5.8e-32):
		tmp = z + (x * 3.0)
	else:
		tmp = z + (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.65e+37) || !(x <= 5.8e-32))
		tmp = Float64(z + Float64(x * 3.0));
	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 <= -3.65e+37) || ~((x <= 5.8e-32)))
		tmp = z + (x * 3.0);
	else
		tmp = z + (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.65e+37], N[Not[LessEqual[x, 5.8e-32]], $MachinePrecision]], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.65000000000000019e37 or 5.79999999999999991e-32 < 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. 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+ 99.8%

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

      12. count-2 99.8%

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

      13. *-commutative 99.8%

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

      14. fma-def 99.8%

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

      15. count-2 99.8%

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

      16. neg-mul-1 99.8%

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

      17. distribute-rgt-out-- 99.8%

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

      18. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 88.5%

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

   if -3.65000000000000019e37 < x < 5.79999999999999991e-32

   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 x around 0 92.0%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.65 \cdot 10^{+37} \lor \neg \left(x \leq 5.8 \cdot 10^{-32}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]

Alternative 6: 78.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+141}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+51}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.6e+141) (* x 3.0) (if (<= x 1.3e+51) (+ z (* y 2.0)) (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e+141) {
		tmp = x * 3.0;
	} else if (x <= 1.3e+51) {
		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 <= (-3.6d+141)) then
        tmp = x * 3.0d0
    else if (x <= 1.3d+51) 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 <= -3.6e+141) {
		tmp = x * 3.0;
	} else if (x <= 1.3e+51) {
		tmp = z + (y * 2.0);
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.6e+141:
		tmp = x * 3.0
	elif x <= 1.3e+51:
		tmp = z + (y * 2.0)
	else:
		tmp = x * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.6e+141)
		tmp = Float64(x * 3.0);
	elseif (x <= 1.3e+51)
		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 <= -3.6e+141)
		tmp = x * 3.0;
	elseif (x <= 1.3e+51)
		tmp = z + (y * 2.0);
	else
		tmp = x * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.6e+141], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, 1.3e+51], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6000000000000001e141 or 1.3000000000000001e51 < 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 73.9%

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

   if -3.6000000000000001e141 < x < 1.3000000000000001e51

   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 85.8%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+141}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+51}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]

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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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 + ((y * 2.0d0) + (x * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative 99.9%

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

   3. *-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 9: 51.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+40}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.9e+40) z (if (<= z 1.5e-58) (* y 2.0) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.9e+40) {
		tmp = z;
	} else if (z <= 1.5e-58) {
		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 <= (-3.9d+40)) then
        tmp = z
    else if (z <= 1.5d-58) 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 <= -3.9e+40) {
		tmp = z;
	} else if (z <= 1.5e-58) {
		tmp = y * 2.0;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.9e+40:
		tmp = z
	elif z <= 1.5e-58:
		tmp = y * 2.0
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.9e+40)
		tmp = z;
	elseif (z <= 1.5e-58)
		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 <= -3.9e+40)
		tmp = z;
	elseif (z <= 1.5e-58)
		tmp = y * 2.0;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.9e+40], z, If[LessEqual[z, 1.5e-58], N[(y * 2.0), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -3.9000000000000001e40 or 1.50000000000000004e-58 < 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 61.8%

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

   if -3.9000000000000001e40 < z < 1.50000000000000004e-58

   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 49.4%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+40}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 10: 34.5% 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 35.6%

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

5. Final simplification 35.6%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023271 
(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))