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

Percentage Accurate: 99.9% → 100.0%

Time: 9.7s

Alternatives: 8

Speedup: 1.0×

• 20.8% 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 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-def 100.0%

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

3. Simplified 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: 60.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+73}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -0.0066:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.14 \cdot 10^{-138}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+97} \lor \neg \left(x \leq 1.16 \cdot 10^{+113}\right) \land \left(x \leq 1.7 \cdot 10^{+170} \lor \neg \left(x \leq 4.2 \cdot 10^{+235}\right) \land x \leq 2.5 \cdot 10^{+258}\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e+73)
   (* z x)
   (if (<= x -0.0066)
     (* x y)
     (if (<= x 1.14e-138)
       (* z 5.0)
       (if (<= x 2.1e+63)
         (* x y)
         (if (or (<= x 8.6e+97)
                 (and (not (<= x 1.16e+113))
                      (or (<= x 1.7e+170)
                          (and (not (<= x 4.2e+235)) (<= x 2.5e+258)))))
           (* z x)
           (* x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e+73) {
		tmp = z * x;
	} else if (x <= -0.0066) {
		tmp = x * y;
	} else if (x <= 1.14e-138) {
		tmp = z * 5.0;
	} else if (x <= 2.1e+63) {
		tmp = x * y;
	} else if ((x <= 8.6e+97) || (!(x <= 1.16e+113) && ((x <= 1.7e+170) || (!(x <= 4.2e+235) && (x <= 2.5e+258))))) {
		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 <= (-1d+73)) then
        tmp = z * x
    else if (x <= (-0.0066d0)) then
        tmp = x * y
    else if (x <= 1.14d-138) then
        tmp = z * 5.0d0
    else if (x <= 2.1d+63) then
        tmp = x * y
    else if ((x <= 8.6d+97) .or. (.not. (x <= 1.16d+113)) .and. (x <= 1.7d+170) .or. (.not. (x <= 4.2d+235)) .and. (x <= 2.5d+258)) 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 <= -1e+73) {
		tmp = z * x;
	} else if (x <= -0.0066) {
		tmp = x * y;
	} else if (x <= 1.14e-138) {
		tmp = z * 5.0;
	} else if (x <= 2.1e+63) {
		tmp = x * y;
	} else if ((x <= 8.6e+97) || (!(x <= 1.16e+113) && ((x <= 1.7e+170) || (!(x <= 4.2e+235) && (x <= 2.5e+258))))) {
		tmp = z * x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e+73:
		tmp = z * x
	elif x <= -0.0066:
		tmp = x * y
	elif x <= 1.14e-138:
		tmp = z * 5.0
	elif x <= 2.1e+63:
		tmp = x * y
	elif (x <= 8.6e+97) or (not (x <= 1.16e+113) and ((x <= 1.7e+170) or (not (x <= 4.2e+235) and (x <= 2.5e+258)))):
		tmp = z * x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e+73)
		tmp = Float64(z * x);
	elseif (x <= -0.0066)
		tmp = Float64(x * y);
	elseif (x <= 1.14e-138)
		tmp = Float64(z * 5.0);
	elseif (x <= 2.1e+63)
		tmp = Float64(x * y);
	elseif ((x <= 8.6e+97) || (!(x <= 1.16e+113) && ((x <= 1.7e+170) || (!(x <= 4.2e+235) && (x <= 2.5e+258)))))
		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 <= -1e+73)
		tmp = z * x;
	elseif (x <= -0.0066)
		tmp = x * y;
	elseif (x <= 1.14e-138)
		tmp = z * 5.0;
	elseif (x <= 2.1e+63)
		tmp = x * y;
	elseif ((x <= 8.6e+97) || (~((x <= 1.16e+113)) && ((x <= 1.7e+170) || (~((x <= 4.2e+235)) && (x <= 2.5e+258)))))
		tmp = z * x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1e+73], N[(z * x), $MachinePrecision], If[LessEqual[x, -0.0066], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.14e-138], N[(z * 5.0), $MachinePrecision], If[LessEqual[x, 2.1e+63], N[(x * y), $MachinePrecision], If[Or[LessEqual[x, 8.6e+97], And[N[Not[LessEqual[x, 1.16e+113]], $MachinePrecision], Or[LessEqual[x, 1.7e+170], And[N[Not[LessEqual[x, 4.2e+235]], $MachinePrecision], LessEqual[x, 2.5e+258]]]]], N[(z * x), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999983e72 or 2.1000000000000002e63 < x < 8.5999999999999996e97 or 1.1600000000000001e113 < x < 1.7000000000000001e170 or 4.2000000000000001e235 < x < 2.5e258

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{1}{z + y}}} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{y + z}}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{y + z}}} \]

   7. Taylor expanded in y around 0 71.8%

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

   if -9.99999999999999983e72 < x < -0.0066 or 1.1399999999999999e-138 < x < 2.1000000000000002e63 or 8.5999999999999996e97 < x < 1.1600000000000001e113 or 1.7000000000000001e170 < x < 4.2000000000000001e235 or 2.5e258 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 69.0%

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

   if -0.0066 < x < 1.1399999999999999e-138

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 77.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+73}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -0.0066:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.14 \cdot 10^{-138}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+97} \lor \neg \left(x \leq 1.16 \cdot 10^{+113}\right) \land \left(x \leq 1.7 \cdot 10^{+170} \lor \neg \left(x \leq 4.2 \cdot 10^{+235}\right) \land x \leq 2.5 \cdot 10^{+258}\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -14 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5 + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -14.0) (not (<= x 5.0))) (* x (+ z y)) (+ (* z 5.0) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -14.0) || !(x <= 5.0)) {
		tmp = x * (z + y);
	} else {
		tmp = (z * 5.0) + (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 <= (-14.0d0)) .or. (.not. (x <= 5.0d0))) then
        tmp = x * (z + y)
    else
        tmp = (z * 5.0d0) + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -14.0) || !(x <= 5.0)) {
		tmp = x * (z + y);
	} else {
		tmp = (z * 5.0) + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -14.0) or not (x <= 5.0):
		tmp = x * (z + y)
	else:
		tmp = (z * 5.0) + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -14.0) || !(x <= 5.0))
		tmp = Float64(x * Float64(z + y));
	else
		tmp = Float64(Float64(z * 5.0) + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -14.0) || ~((x <= 5.0)))
		tmp = x * (z + y);
	else
		tmp = (z * 5.0) + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -14.0], N[Not[LessEqual[x, 5.0]], $MachinePrecision]], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], N[(N[(z * 5.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -14 or 5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.1%

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

      1. +-commutative 99.1%

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

   4. Simplified 99.1%

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

   if -14 < x < 5

   1. Initial program 99.8%

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

      1. flip-+ 55.9%

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

      2. clear-num 55.8%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y - z}{y \cdot y - z \cdot z}}} + z \cdot 5 \]

      3. un-div-inv 55.8%

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

      4. clear-num 55.8%

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{y \cdot y - z \cdot z}{y - z}}}} + z \cdot 5 \]

      5. flip-+ 99.6%

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{y + z}}} + z \cdot 5 \]

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{y + z}}} + z \cdot 5 \]

   4. Taylor expanded in y around inf 98.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5 + x \cdot y\\ \end{array} \]

Alternative 4: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0066 \lor \neg \left(x \leq 5 \cdot 10^{-138}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.0066) (not (<= x 5e-138))) (* x (+ z y)) (* z 5.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.0066) || !(x <= 5e-138)) {
		tmp = x * (z + y);
	} else {
		tmp = z * 5.0;
	}
	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 <= (-0.0066d0)) .or. (.not. (x <= 5d-138))) then
        tmp = x * (z + y)
    else
        tmp = z * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.0066) || !(x <= 5e-138)) {
		tmp = x * (z + y);
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.0066) or not (x <= 5e-138):
		tmp = x * (z + y)
	else:
		tmp = z * 5.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.0066) || !(x <= 5e-138))
		tmp = Float64(x * Float64(z + y));
	else
		tmp = Float64(z * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -0.0066) || ~((x <= 5e-138)))
		tmp = x * (z + y);
	else
		tmp = z * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.0066], N[Not[LessEqual[x, 5e-138]], $MachinePrecision]], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], N[(z * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0066 or 4.99999999999999989e-138 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 93.4%

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

      1. +-commutative 93.4%

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

   4. Simplified 93.4%

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

   if -0.0066 < x < 4.99999999999999989e-138

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 77.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0066 \lor \neg \left(x \leq 5 \cdot 10^{-138}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \]

Alternative 5: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0165 \lor \neg \left(x \leq 4.4 \cdot 10^{-138}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.0165) (not (<= x 4.4e-138))) (* x (+ z y)) (* z (+ 5.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.0165) || !(x <= 4.4e-138)) {
		tmp = x * (z + y);
	} else {
		tmp = z * (5.0 + 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 <= (-0.0165d0)) .or. (.not. (x <= 4.4d-138))) then
        tmp = x * (z + y)
    else
        tmp = z * (5.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.0165) || !(x <= 4.4e-138)) {
		tmp = x * (z + y);
	} else {
		tmp = z * (5.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.0165) or not (x <= 4.4e-138):
		tmp = x * (z + y)
	else:
		tmp = z * (5.0 + x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.0165) || !(x <= 4.4e-138))
		tmp = Float64(x * Float64(z + y));
	else
		tmp = Float64(z * Float64(5.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -0.0165) || ~((x <= 4.4e-138)))
		tmp = x * (z + y);
	else
		tmp = z * (5.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.0165], N[Not[LessEqual[x, 4.4e-138]], $MachinePrecision]], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.016500000000000001 or 4.3999999999999998e-138 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 93.4%

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

      1. +-commutative 93.4%

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

   4. Simplified 93.4%

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

   if -0.016500000000000001 < x < 4.3999999999999998e-138

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 79.7%

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

      1. distribute-rgt-in 79.7%

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

   4. Simplified 79.7%

      \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0165 \lor \neg \left(x \leq 4.4 \cdot 10^{-138}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 7: 60.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0066:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-138}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0066) (* x y) (if (<= x 5.1e-138) (* z 5.0) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0066) {
		tmp = x * y;
	} else if (x <= 5.1e-138) {
		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 <= (-0.0066d0)) then
        tmp = x * y
    else if (x <= 5.1d-138) 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 <= -0.0066) {
		tmp = x * y;
	} else if (x <= 5.1e-138) {
		tmp = z * 5.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.0066:
		tmp = x * y
	elif x <= 5.1e-138:
		tmp = z * 5.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.0066)
		tmp = Float64(x * y);
	elseif (x <= 5.1e-138)
		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 <= -0.0066)
		tmp = x * y;
	elseif (x <= 5.1e-138)
		tmp = z * 5.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.0066], N[(x * y), $MachinePrecision], If[LessEqual[x, 5.1e-138], N[(z * 5.0), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0066 or 5.1000000000000002e-138 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 54.4%

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

   if -0.0066 < x < 5.1000000000000002e-138

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 77.8%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0066:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-138}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 37.6% 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 99.9%

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

2. Taylor expanded in x around 0 32.1%

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

3. Final simplification 32.1%

   \[\leadsto z \cdot 5 \]

Developer target: 97.9% 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 2023297 
(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)))