Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C

Percentage Accurate: 99.9% → 100.0%

Time: 6.3s

Alternatives: 8

Speedup: 1.0×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (y + z)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (y + z)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, 5, x \cdot \left(z + y\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z 5.0 (* x (+ z y))))

C

double code(double x, double y, double z) {
	return fma(z, 5.0, (x * (z + y)));
}

Julia

function code(x, y, z)
	return fma(z, 5.0, Float64(x * Float64(z + y)))
end

Wolfram

code[x_, y_, z_] := N[(z * 5.0 + N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{z \cdot 5 + x \cdot \left(y + z\right)} \]

   2. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(z, 5, x \cdot \left(z + y\right)\right) \]

Alternative 2: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y, z \cdot \left(5 + x\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x y (* z (+ 5.0 x))))

C

double code(double x, double y, double z) {
	return fma(x, y, (z * (5.0 + x)));
}

Julia

function code(x, y, z)
	return fma(x, y, Float64(z * Float64(5.0 + x)))
end

Wolfram

code[x_, y_, z_] := N[(x * y + N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Step-by-step derivation

   1. distribute-lft-in 98.3%

      \[\leadsto \color{blue}{\left(x \cdot y + x \cdot z\right)} + z \cdot 5 \]

   2. associate-+l+ 98.3%

      \[\leadsto \color{blue}{x \cdot y + \left(x \cdot z + z \cdot 5\right)} \]

   3. *-commutative 98.3%

      \[\leadsto x \cdot y + \left(\color{blue}{z \cdot x} + z \cdot 5\right) \]

   4. fma-def 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot x + z \cdot 5\right)} \]

   5. distribute-lft-out 99.5%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(x + 5\right)}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(x + 5\right)\right)} \]

4. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(x, y, z \cdot \left(5 + x\right)\right) \]

Alternative 3: 60.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+203}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+96}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+70}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-95}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+58}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+209}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e+203)
   (* z x)
   (if (<= x -1.4e+96)
     (* x y)
     (if (<= x -6e+70)
       (* z x)
       (if (<= x -4.7e-95)
         (* x y)
         (if (<= x 5.6e-6)
           (* z 5.0)
           (if (<= x 1.06e+58)
             (* x y)
             (if (<= x 9.8e+209) (* z x) (* x y)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+203) {
		tmp = z * x;
	} else if (x <= -1.4e+96) {
		tmp = x * y;
	} else if (x <= -6e+70) {
		tmp = z * x;
	} else if (x <= -4.7e-95) {
		tmp = x * y;
	} else if (x <= 5.6e-6) {
		tmp = z * 5.0;
	} else if (x <= 1.06e+58) {
		tmp = x * y;
	} else if (x <= 9.8e+209) {
		tmp = z * x;
	} else {
		tmp = 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 <= (-3.1d+203)) then
        tmp = z * x
    else if (x <= (-1.4d+96)) then
        tmp = x * y
    else if (x <= (-6d+70)) then
        tmp = z * x
    else if (x <= (-4.7d-95)) then
        tmp = x * y
    else if (x <= 5.6d-6) then
        tmp = z * 5.0d0
    else if (x <= 1.06d+58) then
        tmp = x * y
    else if (x <= 9.8d+209) then
        tmp = z * x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+203) {
		tmp = z * x;
	} else if (x <= -1.4e+96) {
		tmp = x * y;
	} else if (x <= -6e+70) {
		tmp = z * x;
	} else if (x <= -4.7e-95) {
		tmp = x * y;
	} else if (x <= 5.6e-6) {
		tmp = z * 5.0;
	} else if (x <= 1.06e+58) {
		tmp = x * y;
	} else if (x <= 9.8e+209) {
		tmp = z * x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.1e+203:
		tmp = z * x
	elif x <= -1.4e+96:
		tmp = x * y
	elif x <= -6e+70:
		tmp = z * x
	elif x <= -4.7e-95:
		tmp = x * y
	elif x <= 5.6e-6:
		tmp = z * 5.0
	elif x <= 1.06e+58:
		tmp = x * y
	elif x <= 9.8e+209:
		tmp = z * x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.1e+203)
		tmp = Float64(z * x);
	elseif (x <= -1.4e+96)
		tmp = Float64(x * y);
	elseif (x <= -6e+70)
		tmp = Float64(z * x);
	elseif (x <= -4.7e-95)
		tmp = Float64(x * y);
	elseif (x <= 5.6e-6)
		tmp = Float64(z * 5.0);
	elseif (x <= 1.06e+58)
		tmp = Float64(x * y);
	elseif (x <= 9.8e+209)
		tmp = Float64(z * x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.1e+203)
		tmp = z * x;
	elseif (x <= -1.4e+96)
		tmp = x * y;
	elseif (x <= -6e+70)
		tmp = z * x;
	elseif (x <= -4.7e-95)
		tmp = x * y;
	elseif (x <= 5.6e-6)
		tmp = z * 5.0;
	elseif (x <= 1.06e+58)
		tmp = x * y;
	elseif (x <= 9.8e+209)
		tmp = z * x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.1e+203], N[(z * x), $MachinePrecision], If[LessEqual[x, -1.4e+96], N[(x * y), $MachinePrecision], If[LessEqual[x, -6e+70], N[(z * x), $MachinePrecision], If[LessEqual[x, -4.7e-95], N[(x * y), $MachinePrecision], If[LessEqual[x, 5.6e-6], N[(z * 5.0), $MachinePrecision], If[LessEqual[x, 1.06e+58], N[(x * y), $MachinePrecision], If[LessEqual[x, 9.8e+209], N[(z * x), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1e203 or -1.4e96 < x < -5.99999999999999952e70 or 1.05999999999999997e58 < x < 9.7999999999999995e209

   1. Initial program 98.5%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around 0 67.5%

      \[\leadsto \color{blue}{z \cdot x + 5 \cdot z} \]
   3. Step-by-step derivation

      1. +-commutative 67.5%

         \[\leadsto \color{blue}{5 \cdot z + z \cdot x} \]

      2. *-commutative 67.5%

         \[\leadsto 5 \cdot z + \color{blue}{x \cdot z} \]

      3. distribute-rgt-in 68.9%

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   4. Simplified 68.9%

      \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   5. Taylor expanded in x around inf 68.9%

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

   if -3.1e203 < x < -1.4e96 or -5.99999999999999952e70 < x < -4.6999999999999998e-95 or 5.59999999999999975e-6 < x < 1.05999999999999997e58 or 9.7999999999999995e209 < x

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around inf 64.4%

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

   if -4.6999999999999998e-95 < x < 5.59999999999999975e-6

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in x around 0 78.1%

      \[\leadsto \color{blue}{5 \cdot z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+203}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+96}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+70}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-95}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+58}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+209}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -23000 \lor \neg \left(z \leq -2.2 \cdot 10^{-48} \lor \neg \left(z \leq -3.1 \cdot 10^{-86}\right) \land z \leq 5.4 \cdot 10^{-117}\right):\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -23000.0)
         (not
          (or (<= z -2.2e-48) (and (not (<= z -3.1e-86)) (<= z 5.4e-117)))))
   (* z (+ 5.0 x))
   (* x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -23000.0) || !((z <= -2.2e-48) || (!(z <= -3.1e-86) && (z <= 5.4e-117)))) {
		tmp = z * (5.0 + x);
	} else {
		tmp = 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 ((z <= (-23000.0d0)) .or. (.not. (z <= (-2.2d-48)) .or. (.not. (z <= (-3.1d-86))) .and. (z <= 5.4d-117))) then
        tmp = z * (5.0d0 + x)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -23000.0) || !((z <= -2.2e-48) || (!(z <= -3.1e-86) && (z <= 5.4e-117)))) {
		tmp = z * (5.0 + x);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -23000.0) or not ((z <= -2.2e-48) or (not (z <= -3.1e-86) and (z <= 5.4e-117))):
		tmp = z * (5.0 + x)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -23000.0) || !((z <= -2.2e-48) || (!(z <= -3.1e-86) && (z <= 5.4e-117))))
		tmp = Float64(z * Float64(5.0 + x));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -23000.0) || ~(((z <= -2.2e-48) || (~((z <= -3.1e-86)) && (z <= 5.4e-117)))))
		tmp = z * (5.0 + x);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -23000.0], N[Not[Or[LessEqual[z, -2.2e-48], And[N[Not[LessEqual[z, -3.1e-86]], $MachinePrecision], LessEqual[z, 5.4e-117]]]], $MachinePrecision]], N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -23000 or -2.20000000000000013e-48 < z < -3.09999999999999989e-86 or 5.40000000000000005e-117 < z

   1. Initial program 99.2%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around 0 78.9%

      \[\leadsto \color{blue}{z \cdot x + 5 \cdot z} \]
   3. Step-by-step derivation

      1. +-commutative 78.9%

         \[\leadsto \color{blue}{5 \cdot z + z \cdot x} \]

      2. *-commutative 78.9%

         \[\leadsto 5 \cdot z + \color{blue}{x \cdot z} \]

      3. distribute-rgt-in 79.5%

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   4. Simplified 79.5%

      \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   if -23000 < z < -2.20000000000000013e-48 or -3.09999999999999989e-86 < z < 5.40000000000000005e-117

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around inf 70.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -23000 \lor \neg \left(z \leq -2.2 \cdot 10^{-48} \lor \neg \left(z \leq -3.1 \cdot 10^{-86}\right) \land z \leq 5.4 \cdot 10^{-117}\right):\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z + y\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-60}:\\ \;\;\;\;\frac{z}{0.2}\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-95}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ z y))))
   (if (<= x -2.6e-18)
     t_0
     (if (<= x -1.55e-60)
       (/ z 0.2)
       (if (<= x -4.9e-95) (* x y) (if (<= x 6.4e-6) (* z (+ 5.0 x)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z + y);
	double tmp;
	if (x <= -2.6e-18) {
		tmp = t_0;
	} else if (x <= -1.55e-60) {
		tmp = z / 0.2;
	} else if (x <= -4.9e-95) {
		tmp = x * y;
	} else if (x <= 6.4e-6) {
		tmp = z * (5.0 + x);
	} 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 * (z + y)
    if (x <= (-2.6d-18)) then
        tmp = t_0
    else if (x <= (-1.55d-60)) then
        tmp = z / 0.2d0
    else if (x <= (-4.9d-95)) then
        tmp = x * y
    else if (x <= 6.4d-6) then
        tmp = z * (5.0d0 + x)
    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 * (z + y);
	double tmp;
	if (x <= -2.6e-18) {
		tmp = t_0;
	} else if (x <= -1.55e-60) {
		tmp = z / 0.2;
	} else if (x <= -4.9e-95) {
		tmp = x * y;
	} else if (x <= 6.4e-6) {
		tmp = z * (5.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (z + y)
	tmp = 0
	if x <= -2.6e-18:
		tmp = t_0
	elif x <= -1.55e-60:
		tmp = z / 0.2
	elif x <= -4.9e-95:
		tmp = x * y
	elif x <= 6.4e-6:
		tmp = z * (5.0 + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(z + y))
	tmp = 0.0
	if (x <= -2.6e-18)
		tmp = t_0;
	elseif (x <= -1.55e-60)
		tmp = Float64(z / 0.2);
	elseif (x <= -4.9e-95)
		tmp = Float64(x * y);
	elseif (x <= 6.4e-6)
		tmp = Float64(z * Float64(5.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (z + y);
	tmp = 0.0;
	if (x <= -2.6e-18)
		tmp = t_0;
	elseif (x <= -1.55e-60)
		tmp = z / 0.2;
	elseif (x <= -4.9e-95)
		tmp = x * y;
	elseif (x <= 6.4e-6)
		tmp = z * (5.0 + x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6e-18], t$95$0, If[LessEqual[x, -1.55e-60], N[(z / 0.2), $MachinePrecision], If[LessEqual[x, -4.9e-95], N[(x * y), $MachinePrecision], If[LessEqual[x, 6.4e-6], N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.6e-18 or 6.3999999999999997e-6 < x

   1. Initial program 99.3%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in x around inf 98.8%

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

      1. +-commutative 98.8%

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

   4. Simplified 98.8%

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

   if -2.6e-18 < x < -1.54999999999999994e-60

   1. Initial program 99.5%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around 0 70.7%

      \[\leadsto \color{blue}{z \cdot x + 5 \cdot z} \]
   3. Step-by-step derivation

      1. +-commutative 70.7%

         \[\leadsto \color{blue}{5 \cdot z + z \cdot x} \]

      2. *-commutative 70.7%

         \[\leadsto 5 \cdot z + \color{blue}{x \cdot z} \]

      3. distribute-rgt-in 70.7%

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   4. Simplified 70.7%

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

      1. flip-+ 70.7%

         \[\leadsto z \cdot \color{blue}{\frac{5 \cdot 5 - x \cdot x}{5 - x}} \]

      2. associate-*r/ 70.7%

         \[\leadsto \color{blue}{\frac{z \cdot \left(5 \cdot 5 - x \cdot x\right)}{5 - x}} \]

      3. metadata-eval 70.7%

         \[\leadsto \frac{z \cdot \left(\color{blue}{25} - x \cdot x\right)}{5 - x} \]

   6. Applied egg-rr 70.7%

      \[\leadsto \color{blue}{\frac{z \cdot \left(25 - x \cdot x\right)}{5 - x}} \]
   7. Step-by-step derivation

      1. associate-/l* 71.0%

         \[\leadsto \color{blue}{\frac{z}{\frac{5 - x}{25 - x \cdot x}}} \]

   8. Simplified 71.0%

      \[\leadsto \color{blue}{\frac{z}{\frac{5 - x}{25 - x \cdot x}}} \]

   9. Taylor expanded in x around 0 71.0%

      \[\leadsto \frac{z}{\color{blue}{0.2}} \]

   if -1.54999999999999994e-60 < x < -4.9e-95

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around inf 79.2%

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

   if -4.9e-95 < x < 6.3999999999999997e-6

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around 0 78.9%

      \[\leadsto \color{blue}{z \cdot x + 5 \cdot z} \]
   3. Step-by-step derivation

      1. +-commutative 78.9%

         \[\leadsto \color{blue}{5 \cdot z + z \cdot x} \]

      2. *-commutative 78.9%

         \[\leadsto 5 \cdot z + \color{blue}{x \cdot z} \]

      3. distribute-rgt-in 78.9%

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   4. Simplified 78.9%

      \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-60}:\\ \;\;\;\;\frac{z}{0.2}\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-95}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(z + y\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ z y)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (z + y)) + (z * 5.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (z + y)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (z + y)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (z + y)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(z + y)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (z + y)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]

2. Final simplification 99.5%

   \[\leadsto x \cdot \left(z + y\right) + z \cdot 5 \]

Alternative 7: 60.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-95}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-7}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.3e-95) (* x y) (if (<= x 1.56e-7) (* z 5.0) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.3e-95) {
		tmp = x * y;
	} else if (x <= 1.56e-7) {
		tmp = z * 5.0;
	} else {
		tmp = 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 <= (-3.3d-95)) then
        tmp = x * y
    else if (x <= 1.56d-7) then
        tmp = z * 5.0d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.3e-95) {
		tmp = x * y;
	} else if (x <= 1.56e-7) {
		tmp = z * 5.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.3e-95:
		tmp = x * y
	elif x <= 1.56e-7:
		tmp = z * 5.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.3e-95)
		tmp = Float64(x * y);
	elseif (x <= 1.56e-7)
		tmp = Float64(z * 5.0);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.3e-95)
		tmp = x * y;
	elseif (x <= 1.56e-7)
		tmp = z * 5.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.3e-95], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.56e-7], N[(z * 5.0), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3e-95 or 1.55999999999999994e-7 < x

   1. Initial program 99.3%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around inf 53.0%

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

   if -3.3e-95 < x < 1.55999999999999994e-7

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in x around 0 78.1%

      \[\leadsto \color{blue}{5 \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-95}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-7}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 35.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* z 5.0))

C

double code(double x, double y, double z) {
	return z * 5.0;
}

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 * 5.0d0
end function

Java

public static double code(double x, double y, double z) {
	return z * 5.0;
}

Python

def code(x, y, z):
	return z * 5.0

Julia

function code(x, y, z)
	return Float64(z * 5.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = z * 5.0;
end

Wolfram

code[x_, y_, z_] := N[(z * 5.0), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]

2. Taylor expanded in x around 0 33.6%

   \[\leadsto \color{blue}{5 \cdot z} \]

3. Final simplification 33.6%

   \[\leadsto z \cdot 5 \]

Developer target: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + 5\right) \cdot z + x \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (+ x 5.0) z) (* x y)))

C

double code(double x, double y, double z) {
	return ((x + 5.0) * z) + (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 + 5.0d0) * z) + (x * y)
end function

Java

public static double code(double x, double y, double z) {
	return ((x + 5.0) * z) + (x * y);
}

Python

def code(x, y, z):
	return ((x + 5.0) * z) + (x * y)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x + 5.0) * z) + Float64(x * y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x + 5.0) * z) + (x * y);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x + 5.0), $MachinePrecision] * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023224 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :herbie-target
  (+ (* (+ x 5.0) z) (* x y))

  (+ (* x (+ y z)) (* z 5.0)))