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

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Alternatives: 9

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

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 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-+l+ 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

   6. associate-+r+ 99.8%

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

   7. associate-+r+ 99.8%

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

   8. *-lft-identity 99.8%

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

   9. metadata-eval 99.8%

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

   10. count-2 99.8%

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

   11. distribute-rgt-out 99.8%

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

   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. Add Preprocessing

Alternative 2: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+49} \lor \neg \left(x \leq 4.2 \cdot 10^{+26}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2 + \left(z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.2e+49) (not (<= x 4.2e+26)))
   (+ z (* x 3.0))
   (+ (* y 2.0) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.2e+49) || !(x <= 4.2e+26)) {
		tmp = z + (x * 3.0);
	} else {
		tmp = (y * 2.0) + (z + x);
	}
	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 <= (-2.2d+49)) .or. (.not. (x <= 4.2d+26))) then
        tmp = z + (x * 3.0d0)
    else
        tmp = (y * 2.0d0) + (z + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.2e+49) || !(x <= 4.2e+26)) {
		tmp = z + (x * 3.0);
	} else {
		tmp = (y * 2.0) + (z + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.2e+49) or not (x <= 4.2e+26):
		tmp = z + (x * 3.0)
	else:
		tmp = (y * 2.0) + (z + x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.2e+49) || !(x <= 4.2e+26))
		tmp = Float64(z + Float64(x * 3.0));
	else
		tmp = Float64(Float64(y * 2.0) + Float64(z + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.2e+49) || ~((x <= 4.2e+26)))
		tmp = z + (x * 3.0);
	else
		tmp = (y * 2.0) + (z + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.2e+49], N[Not[LessEqual[x, 4.2e+26]], $MachinePrecision]], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * 2.0), $MachinePrecision] + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000001e49 or 4.2000000000000002e26 < x

   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+ 99.7%

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

      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 89.5%

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

   if -2.2000000000000001e49 < x < 4.2000000000000002e26

   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 x around 0 92.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+49} \lor \neg \left(x \leq 4.2 \cdot 10^{+26}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2 + \left(z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+46} \lor \neg \left(x \leq 1.4 \cdot 10^{+28}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e+46) (not (<= x 1.4e+28)))
   (+ z (* x 3.0))
   (+ z (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e+46) || !(x <= 1.4e+28)) {
		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 <= (-1.1d+46)) .or. (.not. (x <= 1.4d+28))) 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 <= -1.1e+46) || !(x <= 1.4e+28)) {
		tmp = z + (x * 3.0);
	} else {
		tmp = z + (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.1e+46) or not (x <= 1.4e+28):
		tmp = z + (x * 3.0)
	else:
		tmp = z + (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.1e+46) || !(x <= 1.4e+28))
		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 <= -1.1e+46) || ~((x <= 1.4e+28)))
		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, -1.1e+46], N[Not[LessEqual[x, 1.4e+28]], $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 < -1.1e46 or 1.4000000000000001e28 < x

   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+ 99.7%

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

      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 89.5%

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

   if -1.1e46 < x < 1.4000000000000001e28

   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 91.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+46} \lor \neg \left(x \leq 1.4 \cdot 10^{+28}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+68} \lor \neg \left(z \leq 1.4 \cdot 10^{-47}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.2e+68) (not (<= z 1.4e-47))) (+ z x) (+ x (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.2e+68) || !(z <= 1.4e-47)) {
		tmp = z + x;
	} else {
		tmp = x + (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 ((z <= (-2.2d+68)) .or. (.not. (z <= 1.4d-47))) then
        tmp = z + x
    else
        tmp = x + (y * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.2e+68) || !(z <= 1.4e-47)) {
		tmp = z + x;
	} else {
		tmp = x + (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.2e+68) or not (z <= 1.4e-47):
		tmp = z + x
	else:
		tmp = x + (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.2e+68) || !(z <= 1.4e-47))
		tmp = Float64(z + x);
	else
		tmp = Float64(x + Float64(y * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.2e+68) || ~((z <= 1.4e-47)))
		tmp = z + x;
	else
		tmp = x + (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.2e+68], N[Not[LessEqual[z, 1.4e-47]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.19999999999999987e68 or 1.39999999999999996e-47 < 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. 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 x around 0 81.0%

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

   6. Taylor expanded in y around 0 68.0%

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

   if -2.19999999999999987e68 < z < 1.39999999999999996e-47

   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 x around 0 63.9%

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

   6. Taylor expanded in z around 0 57.4%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+68} \lor \neg \left(z \leq 1.4 \cdot 10^{-47}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-+l+ 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

   6. associate-+r+ 99.8%

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

   7. associate-+r+ 99.8%

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

   8. *-lft-identity 99.8%

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

   9. metadata-eval 99.8%

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

   10. count-2 99.8%

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

   11. distribute-rgt-out 99.8%

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

   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. Step-by-step derivation

   1. fma-undefine 99.8%

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

   2. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 6: 65.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ z + y \cdot 2 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-+l+ 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

   6. associate-+r+ 99.8%

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

   7. associate-+r+ 99.8%

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

   8. *-lft-identity 99.8%

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

   9. metadata-eval 99.8%

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

   10. count-2 99.8%

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

   11. distribute-rgt-out 99.8%

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

   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 68.8%

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

6. Final simplification 68.8%

   \[\leadsto z + y \cdot 2 \]
7. Add Preprocessing

Alternative 7: 39.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 x around 0 73.3%

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

6. Taylor expanded in y around 0 44.4%

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

7. Final simplification 44.4%

   \[\leadsto z + x \]
8. Add Preprocessing

Alternative 8: 34.2% 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 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-+l+ 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

   6. associate-+r+ 99.8%

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

   7. associate-+r+ 99.8%

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

   8. *-lft-identity 99.8%

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

   9. metadata-eval 99.8%

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

   10. count-2 99.8%

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

   11. distribute-rgt-out 99.8%

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

   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 39.6%

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

Alternative 9: 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 99.8%

   \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
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 x around 0 73.3%

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

6. Taylor expanded in x around inf 7.6%

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

Reproduce

herbie shell --seed 2024163 
(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))