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

Percentage Accurate: 99.9% → 99.9%

Time: 5.9s

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: 99.9% 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. +-commutative 99.9%

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

   9. +-commutative 99.9%

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

   10. associate-+r+ 99.9%

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

   11. associate-+r+ 99.9%

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

   12. associate--l+ 99.9%

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

   13. count-2 99.9%

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

   14. *-commutative 99.9%

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

   15. fma-def 99.9%

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

   16. count-2 99.9%

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

   17. neg-mul-1 99.9%

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

   18. distribute-rgt-out-- 99.9%

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

   19. 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 99.9%

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

   1. +-commutative 99.9%

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

   2. *-commutative 99.9%

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

   3. *-commutative 99.9%

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

   4. fma-def 100.0%

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

   5. *-commutative 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 53.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+42}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-113}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-252}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-210}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+81}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e+42)
   z
   (if (<= z -9.6e-113)
     (* x 3.0)
     (if (<= z -7.5e-289)
       (* 2.0 y)
       (if (<= z 1.95e-252)
         (* x 3.0)
         (if (<= z 6.5e-210) (* 2.0 y) (if (<= z 2.3e+81) (* x 3.0) z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e+42) {
		tmp = z;
	} else if (z <= -9.6e-113) {
		tmp = x * 3.0;
	} else if (z <= -7.5e-289) {
		tmp = 2.0 * y;
	} else if (z <= 1.95e-252) {
		tmp = x * 3.0;
	} else if (z <= 6.5e-210) {
		tmp = 2.0 * y;
	} else if (z <= 2.3e+81) {
		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 <= (-2d+42)) then
        tmp = z
    else if (z <= (-9.6d-113)) then
        tmp = x * 3.0d0
    else if (z <= (-7.5d-289)) then
        tmp = 2.0d0 * y
    else if (z <= 1.95d-252) then
        tmp = x * 3.0d0
    else if (z <= 6.5d-210) then
        tmp = 2.0d0 * y
    else if (z <= 2.3d+81) 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 <= -2e+42) {
		tmp = z;
	} else if (z <= -9.6e-113) {
		tmp = x * 3.0;
	} else if (z <= -7.5e-289) {
		tmp = 2.0 * y;
	} else if (z <= 1.95e-252) {
		tmp = x * 3.0;
	} else if (z <= 6.5e-210) {
		tmp = 2.0 * y;
	} else if (z <= 2.3e+81) {
		tmp = x * 3.0;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2e+42:
		tmp = z
	elif z <= -9.6e-113:
		tmp = x * 3.0
	elif z <= -7.5e-289:
		tmp = 2.0 * y
	elif z <= 1.95e-252:
		tmp = x * 3.0
	elif z <= 6.5e-210:
		tmp = 2.0 * y
	elif z <= 2.3e+81:
		tmp = x * 3.0
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e+42)
		tmp = z;
	elseif (z <= -9.6e-113)
		tmp = Float64(x * 3.0);
	elseif (z <= -7.5e-289)
		tmp = Float64(2.0 * y);
	elseif (z <= 1.95e-252)
		tmp = Float64(x * 3.0);
	elseif (z <= 6.5e-210)
		tmp = Float64(2.0 * y);
	elseif (z <= 2.3e+81)
		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 <= -2e+42)
		tmp = z;
	elseif (z <= -9.6e-113)
		tmp = x * 3.0;
	elseif (z <= -7.5e-289)
		tmp = 2.0 * y;
	elseif (z <= 1.95e-252)
		tmp = x * 3.0;
	elseif (z <= 6.5e-210)
		tmp = 2.0 * y;
	elseif (z <= 2.3e+81)
		tmp = x * 3.0;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e+42], z, If[LessEqual[z, -9.6e-113], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, -7.5e-289], N[(2.0 * y), $MachinePrecision], If[LessEqual[z, 1.95e-252], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, 6.5e-210], N[(2.0 * y), $MachinePrecision], If[LessEqual[z, 2.3e+81], N[(x * 3.0), $MachinePrecision], z]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.00000000000000009e42 or 2.2999999999999999e81 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. associate-+r+ 100.0%

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

      11. associate-+r+ 100.0%

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

      12. associate--l+ 100.0%

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

      13. count-2 100.0%

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

      14. *-commutative 100.0%

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

      15. fma-def 100.0%

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

      16. count-2 100.0%

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

      17. neg-mul-1 100.0%

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

      18. distribute-rgt-out-- 100.0%

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

      19. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 64.2%

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

   if -2.00000000000000009e42 < z < -9.60000000000000049e-113 or -7.49999999999999998e-289 < z < 1.9499999999999999e-252 or 6.49999999999999961e-210 < z < 2.2999999999999999e81

   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. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. associate-+r+ 99.8%

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

      12. associate--l+ 99.8%

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

      13. count-2 99.8%

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

      14. *-commutative 99.8%

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

      15. fma-def 99.8%

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

      16. count-2 99.8%

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

      17. neg-mul-1 99.8%

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

      18. distribute-rgt-out-- 99.8%

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

      19. 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 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. fma-def 99.9%

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

      5. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 57.9%

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

   if -9.60000000000000049e-113 < z < -7.49999999999999998e-289 or 1.9499999999999999e-252 < z < 6.49999999999999961e-210

   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. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. associate-+r+ 100.0%

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

      12. associate--l+ 99.9%

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

      13. count-2 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-def 99.9%

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

      16. count-2 99.9%

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

      17. neg-mul-1 99.9%

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

      18. distribute-rgt-out-- 99.9%

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

      19. 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 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-def 100.0%

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

      5. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 71.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+42}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-113}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-252}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-210}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+81}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + 2 \cdot \left(x + y\right)\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-31}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+56}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* 2.0 (+ x y)))))
   (if (<= y -5.2e+97)
     t_0
     (if (<= y -9e-31)
       (+ z (* 2.0 y))
       (if (<= y 1.1e+56) (+ z (* x 3.0)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (2.0 * (x + y));
	double tmp;
	if (y <= -5.2e+97) {
		tmp = t_0;
	} else if (y <= -9e-31) {
		tmp = z + (2.0 * y);
	} else if (y <= 1.1e+56) {
		tmp = z + (x * 3.0);
	} 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 = x + (2.0d0 * (x + y))
    if (y <= (-5.2d+97)) then
        tmp = t_0
    else if (y <= (-9d-31)) then
        tmp = z + (2.0d0 * y)
    else if (y <= 1.1d+56) then
        tmp = z + (x * 3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (2.0 * (x + y));
	double tmp;
	if (y <= -5.2e+97) {
		tmp = t_0;
	} else if (y <= -9e-31) {
		tmp = z + (2.0 * y);
	} else if (y <= 1.1e+56) {
		tmp = z + (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (2.0 * (x + y))
	tmp = 0
	if y <= -5.2e+97:
		tmp = t_0
	elif y <= -9e-31:
		tmp = z + (2.0 * y)
	elif y <= 1.1e+56:
		tmp = z + (x * 3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(2.0 * Float64(x + y)))
	tmp = 0.0
	if (y <= -5.2e+97)
		tmp = t_0;
	elseif (y <= -9e-31)
		tmp = Float64(z + Float64(2.0 * y));
	elseif (y <= 1.1e+56)
		tmp = Float64(z + Float64(x * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (2.0 * (x + y));
	tmp = 0.0;
	if (y <= -5.2e+97)
		tmp = t_0;
	elseif (y <= -9e-31)
		tmp = z + (2.0 * y);
	elseif (y <= 1.1e+56)
		tmp = z + (x * 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.2e+97], t$95$0, If[LessEqual[y, -9e-31], N[(z + N[(2.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+56], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2e97 or 1.10000000000000008e56 < y

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. count-2 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 89.5%

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

   if -5.2e97 < y < -9.0000000000000008e-31

   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. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. associate-+r+ 99.9%

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

      12. associate--l+ 99.9%

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

      13. count-2 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-def 99.9%

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

      16. count-2 99.9%

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

      17. neg-mul-1 99.9%

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

      18. distribute-rgt-out-- 99.9%

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

      19. 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 inf 89.2%

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

   if -9.0000000000000008e-31 < y < 1.10000000000000008e56

   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. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. associate-+r+ 99.9%

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

      12. associate--l+ 99.9%

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

      13. count-2 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-def 99.9%

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

      16. count-2 99.9%

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

      17. neg-mul-1 99.9%

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

      18. distribute-rgt-out-- 99.9%

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

      19. 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 93.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+97}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-31}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+56}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 4: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+184} \lor \neg \left(x \leq 1.05 \cdot 10^{+165}\right):\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + 2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.05e+184) (not (<= x 1.05e+165))) (* x 3.0) (+ z (* 2.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.05e+184) || !(x <= 1.05e+165)) {
		tmp = x * 3.0;
	} else {
		tmp = z + (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 ((x <= (-1.05d+184)) .or. (.not. (x <= 1.05d+165))) then
        tmp = x * 3.0d0
    else
        tmp = z + (2.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.05e+184) || !(x <= 1.05e+165)) {
		tmp = x * 3.0;
	} else {
		tmp = z + (2.0 * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.05e+184) or not (x <= 1.05e+165):
		tmp = x * 3.0
	else:
		tmp = z + (2.0 * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.05e+184) || !(x <= 1.05e+165))
		tmp = Float64(x * 3.0);
	else
		tmp = Float64(z + Float64(2.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.05e+184) || ~((x <= 1.05e+165)))
		tmp = x * 3.0;
	else
		tmp = z + (2.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.05e+184], N[Not[LessEqual[x, 1.05e+165]], $MachinePrecision]], N[(x * 3.0), $MachinePrecision], N[(z + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e184 or 1.05e165 < 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. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.7%

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

      11. associate-+r+ 99.8%

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

      12. associate--l+ 99.7%

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

      13. count-2 99.7%

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

      14. *-commutative 99.7%

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

      15. fma-def 99.7%

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

      16. count-2 99.7%

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

      17. neg-mul-1 99.7%

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

      18. distribute-rgt-out-- 99.7%

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

      19. 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%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. fma-def 99.8%

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

      5. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 84.2%

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

   if -1.05e184 < x < 1.05e165

   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. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. associate-+r+ 99.9%

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

      12. associate--l+ 99.9%

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

      13. count-2 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-def 99.9%

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

      16. count-2 99.9%

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

      17. neg-mul-1 99.9%

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

      18. distribute-rgt-out-- 99.9%

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

      19. 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 inf 78.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+184} \lor \neg \left(x \leq 1.05 \cdot 10^{+165}\right):\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + 2 \cdot y\\ \end{array} \]

Alternative 5: 85.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-29} \lor \neg \left(y \leq 50\right):\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9.8e-29) (not (<= y 50.0))) (+ z (* 2.0 y)) (+ z (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.8e-29) || !(y <= 50.0)) {
		tmp = z + (2.0 * y);
	} else {
		tmp = z + (x * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-9.8d-29)) .or. (.not. (y <= 50.0d0))) then
        tmp = z + (2.0d0 * y)
    else
        tmp = z + (x * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.8e-29) || !(y <= 50.0)) {
		tmp = z + (2.0 * y);
	} else {
		tmp = z + (x * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -9.8e-29) or not (y <= 50.0):
		tmp = z + (2.0 * y)
	else:
		tmp = z + (x * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -9.8e-29) || !(y <= 50.0))
		tmp = Float64(z + Float64(2.0 * y));
	else
		tmp = Float64(z + Float64(x * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -9.8e-29) || ~((y <= 50.0)))
		tmp = z + (2.0 * y);
	else
		tmp = z + (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -9.8e-29], N[Not[LessEqual[y, 50.0]], $MachinePrecision]], N[(z + N[(2.0 * y), $MachinePrecision]), $MachinePrecision], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.7999999999999997e-29 or 50 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. 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. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. associate-+r+ 100.0%

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

      12. associate--l+ 99.9%

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

      13. count-2 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-def 99.9%

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

      16. count-2 99.9%

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

      17. neg-mul-1 99.9%

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

      18. distribute-rgt-out-- 99.9%

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

      19. 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 inf 80.2%

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

   if -9.7999999999999997e-29 < y < 50

   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. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. associate-+r+ 99.9%

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

      12. associate--l+ 99.9%

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

      13. count-2 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-def 99.9%

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

      16. count-2 99.9%

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

      17. neg-mul-1 99.9%

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

      18. distribute-rgt-out-- 99.9%

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

      19. 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 94.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-29} \lor \neg \left(y \leq 50\right):\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z + N[(N[(2.0 * y), $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. +-commutative 99.9%

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

   9. +-commutative 99.9%

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

   10. associate-+r+ 99.9%

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

   11. associate-+r+ 99.9%

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

   12. associate--l+ 99.9%

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

   13. count-2 99.9%

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

   14. *-commutative 99.9%

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

   15. fma-def 99.9%

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

   16. count-2 99.9%

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

   17. neg-mul-1 99.9%

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

   18. distribute-rgt-out-- 99.9%

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

   19. 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%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 7: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+r+ 99.9%

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

   4. count-2 99.9%

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

   5. +-commutative 99.9%

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

   6. +-commutative 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 8: 53.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+78} \lor \neg \left(y \leq 8.6 \cdot 10^{+55}\right):\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.1e+78) (not (<= y 8.6e+55))) (* 2.0 y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.1e+78) || !(y <= 8.6e+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 ((y <= (-6.1d+78)) .or. (.not. (y <= 8.6d+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 ((y <= -6.1e+78) || !(y <= 8.6e+55)) {
		tmp = 2.0 * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.1e+78) or not (y <= 8.6e+55):
		tmp = 2.0 * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.1e+78) || !(y <= 8.6e+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 ((y <= -6.1e+78) || ~((y <= 8.6e+55)))
		tmp = 2.0 * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.1e+78], N[Not[LessEqual[y, 8.6e+55]], $MachinePrecision]], N[(2.0 * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if y < -6.10000000000000011e78 or 8.5999999999999998e55 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. 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. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. associate-+r+ 100.0%

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

      12. associate--l+ 99.9%

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

      13. count-2 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-def 99.9%

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

      16. count-2 99.9%

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

      17. neg-mul-1 99.9%

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

      18. distribute-rgt-out-- 99.9%

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

      19. 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 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-def 100.0%

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

      5. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 65.4%

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

   if -6.10000000000000011e78 < y < 8.5999999999999998e55

   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. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. associate-+r+ 99.9%

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

      12. associate--l+ 99.9%

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

      13. count-2 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-def 99.9%

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

      16. count-2 99.9%

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

      17. neg-mul-1 99.9%

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

      18. distribute-rgt-out-- 99.9%

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

      19. 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 z around inf 48.8%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+78} \lor \neg \left(y \leq 8.6 \cdot 10^{+55}\right):\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 9: 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%

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

   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. +-commutative 99.9%

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

   9. +-commutative 99.9%

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

   10. associate-+r+ 99.9%

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

   11. associate-+r+ 99.9%

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

   12. associate--l+ 99.9%

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

   13. count-2 99.9%

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

   14. *-commutative 99.9%

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

   15. fma-def 99.9%

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

   16. count-2 99.9%

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

   17. neg-mul-1 99.9%

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

   18. distribute-rgt-out-- 99.9%

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

   19. 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 z around inf 34.4%

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

5. Final simplification 34.4%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023308 
(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))