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

Percentage Accurate: 99.9% → 100.0%

Time: 4.7s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-+l+ 99.9%

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

   3. remove-double-neg 99.9%

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

   4. distribute-neg-in 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. remove-double-neg 99.9%

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

   7. sub-neg 99.9%

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

   8. associate-+l+ 99.9%

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

   9. +-commutative 99.9%

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

   10. associate-+r+ 99.9%

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

   11. associate--l+ 99.9%

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

   12. count-2 99.9%

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

   13. *-commutative 99.9%

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

   14. fma-def 99.9%

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

   15. count-2 99.9%

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

   16. neg-mul-1 99.9%

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

   17. distribute-rgt-out-- 99.9%

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

   18. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. add-log-exp 21.0%

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

   2. *-un-lft-identity 21.0%

      \[\leadsto z + \log \color{blue}{\left(1 \cdot e^{\mathsf{fma}\left(y, 2, x \cdot 3\right)}\right)} \]

   3. log-prod 21.0%

      \[\leadsto z + \color{blue}{\left(\log 1 + \log \left(e^{\mathsf{fma}\left(y, 2, x \cdot 3\right)}\right)\right)} \]

   4. metadata-eval 21.0%

      \[\leadsto z + \left(\color{blue}{0} + \log \left(e^{\mathsf{fma}\left(y, 2, x \cdot 3\right)}\right)\right) \]

   5. add-log-exp 99.9%

      \[\leadsto z + \left(0 + \color{blue}{\mathsf{fma}\left(y, 2, x \cdot 3\right)}\right) \]

   6. fma-udef 99.9%

      \[\leadsto z + \left(0 + \color{blue}{\left(y \cdot 2 + x \cdot 3\right)}\right) \]

   7. +-commutative 99.9%

      \[\leadsto z + \left(0 + \color{blue}{\left(x \cdot 3 + y \cdot 2\right)}\right) \]

   8. fma-def 100.0%

      \[\leadsto z + \left(0 + \color{blue}{\mathsf{fma}\left(x, 3, y \cdot 2\right)}\right) \]

   9. *-commutative 100.0%

      \[\leadsto z + \left(0 + \mathsf{fma}\left(x, 3, \color{blue}{2 \cdot y}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto z + \color{blue}{\left(0 + \mathsf{fma}\left(x, 3, 2 \cdot y\right)\right)} \]

6. Final simplification 100.0%

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

Alternative 2: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+39}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.1e+71)
   (+ z (* x 3.0))
   (if (<= x 3.8e+39) (+ z (* 2.0 y)) (+ x (* 2.0 (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.1e+71) {
		tmp = z + (x * 3.0);
	} else if (x <= 3.8e+39) {
		tmp = z + (2.0 * y);
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.1d+71)) then
        tmp = z + (x * 3.0d0)
    else if (x <= 3.8d+39) then
        tmp = z + (2.0d0 * y)
    else
        tmp = x + (2.0d0 * (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.1e+71) {
		tmp = z + (x * 3.0);
	} else if (x <= 3.8e+39) {
		tmp = z + (2.0 * y);
	} else {
		tmp = x + (2.0 * (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.1e+71:
		tmp = z + (x * 3.0)
	elif x <= 3.8e+39:
		tmp = z + (2.0 * y)
	else:
		tmp = x + (2.0 * (x + y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.1e+71)
		tmp = Float64(z + Float64(x * 3.0));
	elseif (x <= 3.8e+39)
		tmp = Float64(z + Float64(2.0 * y));
	else
		tmp = Float64(x + Float64(2.0 * Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.1e+71)
		tmp = z + (x * 3.0);
	elseif (x <= 3.8e+39)
		tmp = z + (2.0 * y);
	else
		tmp = x + (2.0 * (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.1e+71], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+39], N[(z + N[(2.0 * y), $MachinePrecision]), $MachinePrecision], N[(x + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.09999999999999997e71

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l+ 99.9%

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

      3. remove-double-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-+l+ 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. associate--l+ 99.8%

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

      12. count-2 99.8%

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

      13. *-commutative 99.8%

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

      14. fma-def 99.8%

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

      15. count-2 99.8%

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

      16. neg-mul-1 99.8%

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

      17. distribute-rgt-out-- 99.8%

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

      18. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 87.9%

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

   if -1.09999999999999997e71 < x < 3.7999999999999998e39

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 94.5%

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

   if 3.7999999999999998e39 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. count-2 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 85.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+39}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 3: 52.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+36}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-191}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+112}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.2e+36)
   (* x 3.0)
   (if (<= x -6e-191) z (if (<= x 2.8e+112) (* 2.0 y) (* x 3.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.2e+36) {
		tmp = x * 3.0;
	} else if (x <= -6e-191) {
		tmp = z;
	} else if (x <= 2.8e+112) {
		tmp = 2.0 * y;
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.2d+36)) then
        tmp = x * 3.0d0
    else if (x <= (-6d-191)) then
        tmp = z
    else if (x <= 2.8d+112) then
        tmp = 2.0d0 * y
    else
        tmp = x * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.2e+36) {
		tmp = x * 3.0;
	} else if (x <= -6e-191) {
		tmp = z;
	} else if (x <= 2.8e+112) {
		tmp = 2.0 * y;
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.2e+36:
		tmp = x * 3.0
	elif x <= -6e-191:
		tmp = z
	elif x <= 2.8e+112:
		tmp = 2.0 * y
	else:
		tmp = x * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.2e+36)
		tmp = Float64(x * 3.0);
	elseif (x <= -6e-191)
		tmp = z;
	elseif (x <= 2.8e+112)
		tmp = Float64(2.0 * y);
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.2e+36)
		tmp = x * 3.0;
	elseif (x <= -6e-191)
		tmp = z;
	elseif (x <= 2.8e+112)
		tmp = 2.0 * y;
	else
		tmp = x * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.2e+36], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, -6e-191], z, If[LessEqual[x, 2.8e+112], N[(2.0 * y), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000009e36 or 2.8000000000000001e112 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. count-2 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 66.5%

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

   if -4.20000000000000009e36 < x < -6.0000000000000001e-191

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 61.9%

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

   if -6.0000000000000001e-191 < x < 2.8000000000000001e112

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 53.4%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+36}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-191}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+112}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]

Alternative 4: 84.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+71} \lor \neg \left(x \leq 4.4 \cdot 10^{+107}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + 2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2e+71) (not (<= x 4.4e+107))) (+ z (* x 3.0)) (+ z (* 2.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e+71) || !(x <= 4.4e+107)) {
		tmp = z + (x * 3.0);
	} else {
		tmp = z + (2.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2d+71)) .or. (.not. (x <= 4.4d+107))) then
        tmp = z + (x * 3.0d0)
    else
        tmp = z + (2.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e+71) || !(x <= 4.4e+107)) {
		tmp = z + (x * 3.0);
	} else {
		tmp = z + (2.0 * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2e+71) or not (x <= 4.4e+107):
		tmp = z + (x * 3.0)
	else:
		tmp = z + (2.0 * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2e+71) || !(x <= 4.4e+107))
		tmp = Float64(z + Float64(x * 3.0));
	else
		tmp = Float64(z + Float64(2.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2e+71) || ~((x <= 4.4e+107)))
		tmp = z + (x * 3.0);
	else
		tmp = z + (2.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2e+71], N[Not[LessEqual[x, 4.4e+107]], $MachinePrecision]], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(z + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e71 or 4.4e107 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l+ 99.8%

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

      3. remove-double-neg 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. associate--l+ 99.8%

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

      12. count-2 99.8%

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

      13. *-commutative 99.8%

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

      14. fma-def 99.8%

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

      15. count-2 99.8%

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

      16. neg-mul-1 99.8%

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

      17. distribute-rgt-out-- 99.8%

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

      18. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 84.6%

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

   if -2.0000000000000001e71 < x < 4.4e107

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 90.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+71} \lor \neg \left(x \leq 4.4 \cdot 10^{+107}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + 2 \cdot y\\ \end{array} \]

Alternative 5: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+72}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+150}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.15e+72) (* x 3.0) (if (<= x 9e+150) (+ z (* 2.0 y)) (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e+72) {
		tmp = x * 3.0;
	} else if (x <= 9e+150) {
		tmp = z + (2.0 * y);
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.15d+72)) then
        tmp = x * 3.0d0
    else if (x <= 9d+150) then
        tmp = z + (2.0d0 * y)
    else
        tmp = x * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e+72) {
		tmp = x * 3.0;
	} else if (x <= 9e+150) {
		tmp = z + (2.0 * y);
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.15e+72:
		tmp = x * 3.0
	elif x <= 9e+150:
		tmp = z + (2.0 * y)
	else:
		tmp = x * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.15e+72)
		tmp = Float64(x * 3.0);
	elseif (x <= 9e+150)
		tmp = Float64(z + Float64(2.0 * y));
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.15e+72)
		tmp = x * 3.0;
	elseif (x <= 9e+150)
		tmp = z + (2.0 * y);
	else
		tmp = x * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.15e+72], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, 9e+150], N[(z + N[(2.0 * y), $MachinePrecision]), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e72 or 9.00000000000000001e150 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. count-2 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 71.9%

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

   if -1.15e72 < x < 9.00000000000000001e150

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 88.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+72}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+150}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-+l+ 99.9%

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

   3. +-commutative 99.9%

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

   4. count-2 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 7: 52.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+95}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+136}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.7e+95) z (if (<= z 1.95e+136) (* 2.0 y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.7e+95) {
		tmp = z;
	} else if (z <= 1.95e+136) {
		tmp = 2.0 * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.7d+95)) then
        tmp = z
    else if (z <= 1.95d+136) then
        tmp = 2.0d0 * y
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.7e+95) {
		tmp = z;
	} else if (z <= 1.95e+136) {
		tmp = 2.0 * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.7e+95:
		tmp = z
	elif z <= 1.95e+136:
		tmp = 2.0 * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.7e+95)
		tmp = z;
	elseif (z <= 1.95e+136)
		tmp = Float64(2.0 * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.7e+95)
		tmp = z;
	elseif (z <= 1.95e+136)
		tmp = 2.0 * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.7e+95], z, If[LessEqual[z, 1.95e+136], N[(2.0 * y), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000011e95 or 1.9500000000000001e136 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 71.9%

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

   if -1.70000000000000011e95 < z < 1.9500000000000001e136

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 45.5%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+95}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+136}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 8: 34.9% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-+l+ 99.9%

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

   3. +-commutative 99.9%

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

   4. count-2 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in z around inf 34.0%

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

5. Final simplification 34.0%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023297 
(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))