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

Percentage Accurate: 99.9% → 100.0%

Time: 7.1s

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: 53.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot 2\\ \mathbf{if}\;y \leq -1.26 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-135}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+28}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y 2.0))))
   (if (<= y -1.26e+65)
     t_0
     (if (<= y -3.3e-135) (* x 3.0) (if (<= y 3.7e+28) z t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y * 2.0);
	double tmp;
	if (y <= -1.26e+65) {
		tmp = t_0;
	} else if (y <= -3.3e-135) {
		tmp = x * 3.0;
	} else if (y <= 3.7e+28) {
		tmp = z;
	} 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 + (y * 2.0d0)
    if (y <= (-1.26d+65)) then
        tmp = t_0
    else if (y <= (-3.3d-135)) then
        tmp = x * 3.0d0
    else if (y <= 3.7d+28) then
        tmp = z
    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 + (y * 2.0);
	double tmp;
	if (y <= -1.26e+65) {
		tmp = t_0;
	} else if (y <= -3.3e-135) {
		tmp = x * 3.0;
	} else if (y <= 3.7e+28) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y * 2.0)
	tmp = 0
	if y <= -1.26e+65:
		tmp = t_0
	elif y <= -3.3e-135:
		tmp = x * 3.0
	elif y <= 3.7e+28:
		tmp = z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y * 2.0))
	tmp = 0.0
	if (y <= -1.26e+65)
		tmp = t_0;
	elseif (y <= -3.3e-135)
		tmp = Float64(x * 3.0);
	elseif (y <= 3.7e+28)
		tmp = z;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y * 2.0);
	tmp = 0.0;
	if (y <= -1.26e+65)
		tmp = t_0;
	elseif (y <= -3.3e-135)
		tmp = x * 3.0;
	elseif (y <= 3.7e+28)
		tmp = z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.26e+65], t$95$0, If[LessEqual[y, -3.3e-135], N[(x * 3.0), $MachinePrecision], If[LessEqual[y, 3.7e+28], z, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2599999999999999e65 or 3.6999999999999999e28 < 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. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in z around 0 84.4%

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

   6. Taylor expanded in x around 0 72.6%

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

   if -1.2599999999999999e65 < y < -3.2999999999999999e-135

   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. associate-+l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. count-2 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 67.3%

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

   6. Taylor expanded in x around inf 55.6%

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

      1. *-commutative 55.6%

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

   8. Simplified 55.6%

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

   if -3.2999999999999999e-135 < y < 3.6999999999999999e28

   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. +-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 54.8%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+65}:\\ \;\;\;\;x + y \cdot 2\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-135}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+28}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+66}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-134}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+34}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.6e+66)
   (* y 2.0)
   (if (<= y -5.5e-134) (* x 3.0) (if (<= y 5.6e+34) z (* y 2.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6e+66) {
		tmp = y * 2.0;
	} else if (y <= -5.5e-134) {
		tmp = x * 3.0;
	} else if (y <= 5.6e+34) {
		tmp = z;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.6d+66)) then
        tmp = y * 2.0d0
    else if (y <= (-5.5d-134)) then
        tmp = x * 3.0d0
    else if (y <= 5.6d+34) then
        tmp = z
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6e+66) {
		tmp = y * 2.0;
	} else if (y <= -5.5e-134) {
		tmp = x * 3.0;
	} else if (y <= 5.6e+34) {
		tmp = z;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.6e+66:
		tmp = y * 2.0
	elif y <= -5.5e-134:
		tmp = x * 3.0
	elif y <= 5.6e+34:
		tmp = z
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.6e+66)
		tmp = Float64(y * 2.0);
	elseif (y <= -5.5e-134)
		tmp = Float64(x * 3.0);
	elseif (y <= 5.6e+34)
		tmp = z;
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.6e+66)
		tmp = y * 2.0;
	elseif (y <= -5.5e-134)
		tmp = x * 3.0;
	elseif (y <= 5.6e+34)
		tmp = z;
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.6e+66], N[(y * 2.0), $MachinePrecision], If[LessEqual[y, -5.5e-134], N[(x * 3.0), $MachinePrecision], If[LessEqual[y, 5.6e+34], z, N[(y * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6e66 or 5.60000000000000016e34 < 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. +-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. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+l+ 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+r+ 100.0%

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

      8. *-lft-identity 100.0%

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

      9. metadata-eval 100.0%

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

      10. count-2 100.0%

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

      11. distribute-rgt-out 100.0%

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

      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 86.0%

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

   6. Taylor expanded in z around 0 70.9%

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

   if -4.6e66 < y < -5.5000000000000002e-134

   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. associate-+l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. count-2 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 67.3%

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

   6. Taylor expanded in x around inf 55.6%

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

      1. *-commutative 55.6%

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

   8. Simplified 55.6%

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

   if -5.5000000000000002e-134 < y < 5.60000000000000016e34

   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. +-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 54.8%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+66}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-134}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+34}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+64} \lor \neg \left(y \leq 2.25 \cdot 10^{+21}\right):\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.5e+64) (not (<= y 2.25e+21)))
   (+ z (* y 2.0))
   (+ z (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.5e+64) || !(y <= 2.25e+21)) {
		tmp = z + (y * 2.0);
	} else {
		tmp = z + (x * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-7.5d+64)) .or. (.not. (y <= 2.25d+21))) then
        tmp = z + (y * 2.0d0)
    else
        tmp = z + (x * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.5e+64) || !(y <= 2.25e+21)) {
		tmp = z + (y * 2.0);
	} else {
		tmp = z + (x * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.5e+64) or not (y <= 2.25e+21):
		tmp = z + (y * 2.0)
	else:
		tmp = z + (x * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.5e+64) || !(y <= 2.25e+21))
		tmp = Float64(z + Float64(y * 2.0));
	else
		tmp = Float64(z + Float64(x * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7.5e+64) || ~((y <= 2.25e+21)))
		tmp = z + (y * 2.0);
	else
		tmp = z + (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.5e+64], N[Not[LessEqual[y, 2.25e+21]], $MachinePrecision]], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5000000000000005e64 or 2.25e21 < 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. +-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. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+l+ 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+r+ 100.0%

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

      8. *-lft-identity 100.0%

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

      9. metadata-eval 100.0%

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

      10. count-2 100.0%

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

      11. distribute-rgt-out 100.0%

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

      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 86.4%

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

   if -7.5000000000000005e64 < y < 2.25e21

   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. +-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 x around inf 93.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+64} \lor \neg \left(y \leq 2.25 \cdot 10^{+21}\right):\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+238} \lor \neg \left(x \leq 7.6 \cdot 10^{+225}\right):\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.15e+238) (not (<= x 7.6e+225))) (* x 3.0) (+ z (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.15e+238) || !(x <= 7.6e+225)) {
		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 <= (-1.15d+238)) .or. (.not. (x <= 7.6d+225))) 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 <= -1.15e+238) || !(x <= 7.6e+225)) {
		tmp = x * 3.0;
	} else {
		tmp = z + (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.15e+238) or not (x <= 7.6e+225):
		tmp = x * 3.0
	else:
		tmp = z + (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.15e+238) || !(x <= 7.6e+225))
		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 <= -1.15e+238) || ~((x <= 7.6e+225)))
		tmp = x * 3.0;
	else
		tmp = z + (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.15e+238], N[Not[LessEqual[x, 7.6e+225]], $MachinePrecision]], N[(x * 3.0), $MachinePrecision], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15000000000000001e238 or 7.6000000000000002e225 < 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. associate-+l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. count-2 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 93.8%

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

   6. Taylor expanded in x around inf 90.7%

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

      1. *-commutative 90.7%

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

   8. Simplified 90.7%

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

   if -1.15000000000000001e238 < x < 7.6000000000000002e225

   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 77.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+238} \lor \neg \left(x \leq 7.6 \cdot 10^{+225}\right):\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+67}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+21}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.65e+67)
   (+ x (* 2.0 (+ x y)))
   (if (<= y 6.4e+21) (+ z (* x 3.0)) (+ z (* y 2.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.65e+67) {
		tmp = x + (2.0 * (x + y));
	} else if (y <= 6.4e+21) {
		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 (y <= (-2.65d+67)) then
        tmp = x + (2.0d0 * (x + y))
    else if (y <= 6.4d+21) 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 (y <= -2.65e+67) {
		tmp = x + (2.0 * (x + y));
	} else if (y <= 6.4e+21) {
		tmp = z + (x * 3.0);
	} else {
		tmp = z + (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.65e+67:
		tmp = x + (2.0 * (x + y))
	elif y <= 6.4e+21:
		tmp = z + (x * 3.0)
	else:
		tmp = z + (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.65e+67)
		tmp = Float64(x + Float64(2.0 * Float64(x + y)));
	elseif (y <= 6.4e+21)
		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 (y <= -2.65e+67)
		tmp = x + (2.0 * (x + y));
	elseif (y <= 6.4e+21)
		tmp = z + (x * 3.0);
	else
		tmp = z + (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.65e+67], N[(x + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.4e+21], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(z + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.65e67

   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. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in z around 0 87.1%

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

   if -2.65e67 < y < 6.4e21

   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. +-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 x around inf 93.0%

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

   if 6.4e21 < 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. +-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. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+l+ 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+r+ 100.0%

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

      8. *-lft-identity 100.0%

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

      9. metadata-eval 100.0%

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

      10. count-2 100.0%

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

      11. distribute-rgt-out 100.0%

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

      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 91.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+67}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+21}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+66} \lor \neg \left(y \leq 2.6 \cdot 10^{+45}\right):\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.85e+66) (not (<= y 2.6e+45))) (* y 2.0) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.85e+66) || !(y <= 2.6e+45)) {
		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 ((y <= (-1.85d+66)) .or. (.not. (y <= 2.6d+45))) 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 ((y <= -1.85e+66) || !(y <= 2.6e+45)) {
		tmp = y * 2.0;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.85e+66) or not (y <= 2.6e+45):
		tmp = y * 2.0
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.85e+66) || !(y <= 2.6e+45))
		tmp = Float64(y * 2.0);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.85e+66) || ~((y <= 2.6e+45)))
		tmp = y * 2.0;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.85e+66], N[Not[LessEqual[y, 2.6e+45]], $MachinePrecision]], N[(y * 2.0), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if y < -1.85e66 or 2.60000000000000007e45 < 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. +-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. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+l+ 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+r+ 100.0%

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

      8. *-lft-identity 100.0%

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

      9. metadata-eval 100.0%

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

      10. count-2 100.0%

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

      11. distribute-rgt-out 100.0%

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

      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 86.0%

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

   6. Taylor expanded in z around 0 70.9%

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

   if -1.85e66 < y < 2.60000000000000007e45

   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. +-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 50.8%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+66} \lor \neg \left(y \leq 2.6 \cdot 10^{+45}\right):\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z + N[(N[(x * 3.0), $MachinePrecision] + 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. 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. Final simplification 99.9%

   \[\leadsto z + \left(x \cdot 3 + y \cdot 2\right) \]
8. Add Preprocessing

Alternative 9: 34.3% 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 37.9%

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

Alternative 10: 8.0% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

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. Taylor expanded in z around 0 63.3%

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

6. Taylor expanded in x around 0 37.6%

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

7. Taylor expanded in x around inf 7.6%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(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))