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

Percentage Accurate: 99.9% → 100.0%

Time: 7.8s

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: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-+l+ 99.9%

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

   3. +-commutative 99.9%

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

   4. +-commutative 99.9%

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

   5. associate-+l+ 99.9%

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

   6. associate-+r+ 99.9%

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

   7. associate-+r+ 99.9%

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

   8. *-lft-identity 99.9%

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

   9. metadata-eval 99.9%

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

   10. count-2 99.9%

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

   11. distribute-rgt-out 99.9%

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

   12. fma-define 99.9%

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

   13. metadata-eval 99.9%

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

   14. metadata-eval 99.9%

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

   15. count-2 99.9%

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

   16. *-commutative 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 52.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+17}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-208}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-295}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.4e+17)
   z
   (if (<= z -1.6e-208)
     (* y 2.0)
     (if (<= z 9e-295) (* x 3.0) (if (<= z 4.3e-23) (* y 2.0) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.4e+17) {
		tmp = z;
	} else if (z <= -1.6e-208) {
		tmp = y * 2.0;
	} else if (z <= 9e-295) {
		tmp = x * 3.0;
	} else if (z <= 4.3e-23) {
		tmp = y * 2.0;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.4d+17)) then
        tmp = z
    else if (z <= (-1.6d-208)) then
        tmp = y * 2.0d0
    else if (z <= 9d-295) then
        tmp = x * 3.0d0
    else if (z <= 4.3d-23) then
        tmp = y * 2.0d0
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.4e+17) {
		tmp = z;
	} else if (z <= -1.6e-208) {
		tmp = y * 2.0;
	} else if (z <= 9e-295) {
		tmp = x * 3.0;
	} else if (z <= 4.3e-23) {
		tmp = y * 2.0;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.4e+17:
		tmp = z
	elif z <= -1.6e-208:
		tmp = y * 2.0
	elif z <= 9e-295:
		tmp = x * 3.0
	elif z <= 4.3e-23:
		tmp = y * 2.0
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.4e+17)
		tmp = z;
	elseif (z <= -1.6e-208)
		tmp = Float64(y * 2.0);
	elseif (z <= 9e-295)
		tmp = Float64(x * 3.0);
	elseif (z <= 4.3e-23)
		tmp = Float64(y * 2.0);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.4e+17)
		tmp = z;
	elseif (z <= -1.6e-208)
		tmp = y * 2.0;
	elseif (z <= 9e-295)
		tmp = x * 3.0;
	elseif (z <= 4.3e-23)
		tmp = y * 2.0;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.4e+17], z, If[LessEqual[z, -1.6e-208], N[(y * 2.0), $MachinePrecision], If[LessEqual[z, 9e-295], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, 4.3e-23], N[(y * 2.0), $MachinePrecision], z]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4e17 or 4.30000000000000002e-23 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. associate-+r+ 99.9%

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

      8. *-lft-identity 99.9%

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

      9. metadata-eval 99.9%

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

      10. count-2 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. fma-define 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. count-2 99.9%

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

      16. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 62.2%

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

   if -2.4e17 < z < -1.6000000000000001e-208 or 9.0000000000000003e-295 < z < 4.30000000000000002e-23

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. associate-+r+ 99.9%

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

      8. *-lft-identity 99.9%

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

      9. metadata-eval 99.9%

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

      10. count-2 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. fma-define 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. count-2 100.0%

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

      16. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 90.9%

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

   6. Taylor expanded in y around inf 57.0%

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

   if -1.6000000000000001e-208 < z < 9.0000000000000003e-295

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+l+ 99.6%

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

      6. associate-+r+ 99.6%

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

      7. associate-+r+ 99.6%

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

      8. *-lft-identity 99.6%

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

      9. metadata-eval 99.6%

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

      10. count-2 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. fma-define 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. count-2 99.9%

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

      16. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 80.2%

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

   6. Taylor expanded in x around inf 80.2%

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

   7. Taylor expanded in y around 0 69.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+17}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-208}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-295}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+95}:\\ \;\;\;\;\left(z + x\right) + x \cdot 2\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+88}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.5e+95)
   (+ (+ z x) (* x 2.0))
   (if (<= x 6.6e+88) (+ z (* y 2.0)) (+ x (* 2.0 (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.5e+95) {
		tmp = (z + x) + (x * 2.0);
	} else if (x <= 6.6e+88) {
		tmp = z + (y * 2.0);
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.5d+95)) then
        tmp = (z + x) + (x * 2.0d0)
    else if (x <= 6.6d+88) then
        tmp = z + (y * 2.0d0)
    else
        tmp = x + (2.0d0 * (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.5e+95) {
		tmp = (z + x) + (x * 2.0);
	} else if (x <= 6.6e+88) {
		tmp = z + (y * 2.0);
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.5e+95:
		tmp = (z + x) + (x * 2.0)
	elif x <= 6.6e+88:
		tmp = z + (y * 2.0)
	else:
		tmp = x + (2.0 * (x + y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.5e+95)
		tmp = Float64(Float64(z + x) + Float64(x * 2.0));
	elseif (x <= 6.6e+88)
		tmp = Float64(z + Float64(y * 2.0));
	else
		tmp = Float64(x + Float64(2.0 * Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.5e+95)
		tmp = (z + x) + (x * 2.0);
	elseif (x <= 6.6e+88)
		tmp = z + (y * 2.0);
	else
		tmp = x + (2.0 * (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.5e+95], N[(N[(z + x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e+88], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.50000000000000017e95

   1. Initial program 99.6%

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

      1. associate-+l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. count-2 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 85.8%

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

   if -4.50000000000000017e95 < x < 6.6000000000000006e88

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. associate-+r+ 99.9%

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

      8. *-lft-identity 99.9%

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

      9. metadata-eval 99.9%

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

      10. count-2 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. fma-define 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. count-2 100.0%

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

      16. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 89.8%

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

   if 6.6000000000000006e88 < 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. associate-+l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. count-2 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 86.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+95}:\\ \;\;\;\;\left(z + x\right) + x \cdot 2\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+88}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+94}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+87}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.5e+94)
   (+ z (* x 3.0))
   (if (<= x 7.5e+87) (+ z (* y 2.0)) (+ x (* 2.0 (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.5e+94) {
		tmp = z + (x * 3.0);
	} else if (x <= 7.5e+87) {
		tmp = z + (y * 2.0);
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-6.5d+94)) then
        tmp = z + (x * 3.0d0)
    else if (x <= 7.5d+87) then
        tmp = z + (y * 2.0d0)
    else
        tmp = x + (2.0d0 * (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.5e+94) {
		tmp = z + (x * 3.0);
	} else if (x <= 7.5e+87) {
		tmp = z + (y * 2.0);
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.5e+94:
		tmp = z + (x * 3.0)
	elif x <= 7.5e+87:
		tmp = z + (y * 2.0)
	else:
		tmp = x + (2.0 * (x + y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.5e+94)
		tmp = Float64(z + Float64(x * 3.0));
	elseif (x <= 7.5e+87)
		tmp = Float64(z + Float64(y * 2.0));
	else
		tmp = Float64(x + Float64(2.0 * Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.5e+94)
		tmp = z + (x * 3.0);
	elseif (x <= 7.5e+87)
		tmp = z + (y * 2.0);
	else
		tmp = x + (2.0 * (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.5e+94], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+87], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.49999999999999976e94

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+l+ 99.7%

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

      6. associate-+r+ 99.7%

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

      7. associate-+r+ 99.7%

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

      8. *-lft-identity 99.7%

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

      9. metadata-eval 99.7%

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

      10. count-2 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. fma-define 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. count-2 99.9%

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

      16. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 85.7%

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

   if -6.49999999999999976e94 < x < 7.50000000000000014e87

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. associate-+r+ 99.9%

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

      8. *-lft-identity 99.9%

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

      9. metadata-eval 99.9%

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

      10. count-2 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. fma-define 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. count-2 100.0%

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

      16. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 89.8%

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

   if 7.50000000000000014e87 < 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. associate-+l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. count-2 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 86.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+94}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+87}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+95} \lor \neg \left(x \leq 4.1 \cdot 10^{+66}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.7e+95) (not (<= x 4.1e+66)))
   (+ z (* x 3.0))
   (+ z (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e+95) || !(x <= 4.1e+66)) {
		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 <= (-3.7d+95)) .or. (.not. (x <= 4.1d+66))) 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 <= -3.7e+95) || !(x <= 4.1e+66)) {
		tmp = z + (x * 3.0);
	} else {
		tmp = z + (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.7e+95) or not (x <= 4.1e+66):
		tmp = z + (x * 3.0)
	else:
		tmp = z + (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.7e+95) || !(x <= 4.1e+66))
		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 <= -3.7e+95) || ~((x <= 4.1e+66)))
		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, -3.7e+95], N[Not[LessEqual[x, 4.1e+66]], $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 < -3.7000000000000001e95 or 4.09999999999999994e66 < 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+ 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+l+ 99.7%

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

      6. associate-+r+ 99.7%

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

      7. associate-+r+ 99.7%

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

      8. *-lft-identity 99.7%

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

      9. metadata-eval 99.7%

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

      10. count-2 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. fma-define 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. count-2 99.8%

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

      16. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 82.4%

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

   if -3.7000000000000001e95 < x < 4.09999999999999994e66

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. associate-+r+ 99.9%

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

      8. *-lft-identity 99.9%

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

      9. metadata-eval 99.9%

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

      10. count-2 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. fma-define 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. count-2 100.0%

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

      16. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 90.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+95} \lor \neg \left(x \leq 4.1 \cdot 10^{+66}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+110} \lor \neg \left(x \leq 2.9 \cdot 10^{+176}\right):\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.8e+110) (not (<= x 2.9e+176))) (* x 3.0) (+ z (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.8e+110) || !(x <= 2.9e+176)) {
		tmp = 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 <= (-2.8d+110)) .or. (.not. (x <= 2.9d+176))) then
        tmp = 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 <= -2.8e+110) || !(x <= 2.9e+176)) {
		tmp = x * 3.0;
	} else {
		tmp = z + (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.8e+110) or not (x <= 2.9e+176):
		tmp = x * 3.0
	else:
		tmp = z + (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.8e+110) || !(x <= 2.9e+176))
		tmp = 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 <= -2.8e+110) || ~((x <= 2.9e+176)))
		tmp = x * 3.0;
	else
		tmp = z + (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.8e+110], N[Not[LessEqual[x, 2.9e+176]], $MachinePrecision]], N[(x * 3.0), $MachinePrecision], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.79999999999999987e110 or 2.9000000000000001e176 < 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+ 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+l+ 99.7%

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

      6. associate-+r+ 99.7%

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

      7. associate-+r+ 99.7%

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

      8. *-lft-identity 99.7%

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

      9. metadata-eval 99.7%

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

      10. count-2 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. fma-define 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. count-2 99.8%

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

      16. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 67.5%

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

   6. Taylor expanded in x around inf 61.8%

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

   7. Taylor expanded in y around 0 74.0%

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

   if -2.79999999999999987e110 < x < 2.9000000000000001e176

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. associate-+r+ 99.9%

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

      8. *-lft-identity 99.9%

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

      9. metadata-eval 99.9%

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

      10. count-2 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. fma-define 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. count-2 100.0%

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

      16. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 86.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+110} \lor \neg \left(x \leq 2.9 \cdot 10^{+176}\right):\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+95} \lor \neg \left(x \leq 7.4 \cdot 10^{+86}\right):\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.12e+95) (not (<= x 7.4e+86))) (* x 3.0) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.12e+95) || !(x <= 7.4e+86)) {
		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 ((x <= (-1.12d+95)) .or. (.not. (x <= 7.4d+86))) 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 ((x <= -1.12e+95) || !(x <= 7.4e+86)) {
		tmp = x * 3.0;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.12e+95) or not (x <= 7.4e+86):
		tmp = x * 3.0
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.12e+95) || !(x <= 7.4e+86))
		tmp = Float64(x * 3.0);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.12e+95) || ~((x <= 7.4e+86)))
		tmp = x * 3.0;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.12e+95], N[Not[LessEqual[x, 7.4e+86]], $MachinePrecision]], N[(x * 3.0), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -1.11999999999999999e95 or 7.39999999999999983e86 < 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+ 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+l+ 99.7%

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

      6. associate-+r+ 99.7%

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

      7. associate-+r+ 99.7%

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

      8. *-lft-identity 99.7%

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

      9. metadata-eval 99.7%

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

      10. count-2 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. fma-define 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. count-2 99.8%

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

      16. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 69.1%

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

   6. Taylor expanded in x around inf 60.6%

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

   7. Taylor expanded in y around 0 68.5%

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

   if -1.11999999999999999e95 < x < 7.39999999999999983e86

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. associate-+r+ 99.9%

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

      8. *-lft-identity 99.9%

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

      9. metadata-eval 99.9%

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

      10. count-2 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. fma-define 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. count-2 100.0%

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

      16. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 52.4%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+95} \lor \neg \left(x \leq 7.4 \cdot 10^{+86}\right):\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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. associate-+l+ 99.9%

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

   3. +-commutative 99.9%

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

   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. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto 2 \cdot \left(x + y\right) + \left(z + x\right) \]
6. Add Preprocessing

Alternative 9: 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. +-commutative 99.9%

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

   4. +-commutative 99.9%

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

   5. associate-+l+ 99.9%

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

   6. associate-+r+ 99.9%

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

   7. associate-+r+ 99.9%

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

   8. *-lft-identity 99.9%

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

   9. metadata-eval 99.9%

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

   10. count-2 99.9%

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

   11. distribute-rgt-out 99.9%

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

   12. fma-define 99.9%

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

   13. metadata-eval 99.9%

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

   14. metadata-eval 99.9%

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

   15. count-2 99.9%

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

   16. *-commutative 99.9%

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

3. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

   2. +-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 10: 33.9% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-+l+ 99.9%

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

   3. +-commutative 99.9%

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

   4. +-commutative 99.9%

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

   5. associate-+l+ 99.9%

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

   6. associate-+r+ 99.9%

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

   7. associate-+r+ 99.9%

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

   8. *-lft-identity 99.9%

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

   9. metadata-eval 99.9%

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

   10. count-2 99.9%

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

   11. distribute-rgt-out 99.9%

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

   12. fma-define 99.9%

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

   13. metadata-eval 99.9%

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

   14. metadata-eval 99.9%

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

   15. count-2 99.9%

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

   16. *-commutative 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in z around inf 40.0%

   \[\leadsto \color{blue}{z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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))