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

Percentage Accurate: 99.9% → 100.0%

Time: 6.1s

Alternatives: 8

Speedup: 1.0×

• 20.9% 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. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 61.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+250}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+77}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+15}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-29}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+29} \lor \neg \left(x \leq 1.18 \cdot 10^{+101}\right) \land \left(x \leq 8.4 \cdot 10^{+133} \lor \neg \left(x \leq 5.8 \cdot 10^{+163}\right) \land x \leq 2 \cdot 10^{+235}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.4e+250)
   (* z x)
   (if (<= x -2.8e+77)
     (* x y)
     (if (<= x -2e+15)
       (* z x)
       (if (<= x 9.8e-29)
         (* z 5.0)
         (if (or (<= x 5.8e+29)
                 (and (not (<= x 1.18e+101))
                      (or (<= x 8.4e+133)
                          (and (not (<= x 5.8e+163)) (<= x 2e+235)))))
           (* x y)
           (* z x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e+250) {
		tmp = z * x;
	} else if (x <= -2.8e+77) {
		tmp = x * y;
	} else if (x <= -2e+15) {
		tmp = z * x;
	} else if (x <= 9.8e-29) {
		tmp = z * 5.0;
	} else if ((x <= 5.8e+29) || (!(x <= 1.18e+101) && ((x <= 8.4e+133) || (!(x <= 5.8e+163) && (x <= 2e+235))))) {
		tmp = x * y;
	} else {
		tmp = 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 <= (-3.4d+250)) then
        tmp = z * x
    else if (x <= (-2.8d+77)) then
        tmp = x * y
    else if (x <= (-2d+15)) then
        tmp = z * x
    else if (x <= 9.8d-29) then
        tmp = z * 5.0d0
    else if ((x <= 5.8d+29) .or. (.not. (x <= 1.18d+101)) .and. (x <= 8.4d+133) .or. (.not. (x <= 5.8d+163)) .and. (x <= 2d+235)) then
        tmp = x * y
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e+250) {
		tmp = z * x;
	} else if (x <= -2.8e+77) {
		tmp = x * y;
	} else if (x <= -2e+15) {
		tmp = z * x;
	} else if (x <= 9.8e-29) {
		tmp = z * 5.0;
	} else if ((x <= 5.8e+29) || (!(x <= 1.18e+101) && ((x <= 8.4e+133) || (!(x <= 5.8e+163) && (x <= 2e+235))))) {
		tmp = x * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.4e+250:
		tmp = z * x
	elif x <= -2.8e+77:
		tmp = x * y
	elif x <= -2e+15:
		tmp = z * x
	elif x <= 9.8e-29:
		tmp = z * 5.0
	elif (x <= 5.8e+29) or (not (x <= 1.18e+101) and ((x <= 8.4e+133) or (not (x <= 5.8e+163) and (x <= 2e+235)))):
		tmp = x * y
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.4e+250)
		tmp = Float64(z * x);
	elseif (x <= -2.8e+77)
		tmp = Float64(x * y);
	elseif (x <= -2e+15)
		tmp = Float64(z * x);
	elseif (x <= 9.8e-29)
		tmp = Float64(z * 5.0);
	elseif ((x <= 5.8e+29) || (!(x <= 1.18e+101) && ((x <= 8.4e+133) || (!(x <= 5.8e+163) && (x <= 2e+235)))))
		tmp = Float64(x * y);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.4e+250)
		tmp = z * x;
	elseif (x <= -2.8e+77)
		tmp = x * y;
	elseif (x <= -2e+15)
		tmp = z * x;
	elseif (x <= 9.8e-29)
		tmp = z * 5.0;
	elseif ((x <= 5.8e+29) || (~((x <= 1.18e+101)) && ((x <= 8.4e+133) || (~((x <= 5.8e+163)) && (x <= 2e+235)))))
		tmp = x * y;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.4e+250], N[(z * x), $MachinePrecision], If[LessEqual[x, -2.8e+77], N[(x * y), $MachinePrecision], If[LessEqual[x, -2e+15], N[(z * x), $MachinePrecision], If[LessEqual[x, 9.8e-29], N[(z * 5.0), $MachinePrecision], If[Or[LessEqual[x, 5.8e+29], And[N[Not[LessEqual[x, 1.18e+101]], $MachinePrecision], Or[LessEqual[x, 8.4e+133], And[N[Not[LessEqual[x, 5.8e+163]], $MachinePrecision], LessEqual[x, 2e+235]]]]], N[(x * y), $MachinePrecision], N[(z * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.39999999999999973e250 or -2.8e77 < x < -2e15 or 5.7999999999999999e29 < x < 1.18000000000000005e101 or 8.4e133 < x < 5.79999999999999996e163 or 2.0000000000000001e235 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 73.9%

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

   if -3.39999999999999973e250 < x < -2.8e77 or 9.7999999999999997e-29 < x < 5.7999999999999999e29 or 1.18000000000000005e101 < x < 8.4e133 or 5.79999999999999996e163 < x < 2.0000000000000001e235

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 72.5%

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

   if -2e15 < x < 9.7999999999999997e-29

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 71.0%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+250}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+77}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+15}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-29}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+29} \lor \neg \left(x \leq 1.18 \cdot 10^{+101}\right) \land \left(x \leq 8.4 \cdot 10^{+133} \lor \neg \left(x \leq 5.8 \cdot 10^{+163}\right) \land x \leq 2 \cdot 10^{+235}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z + y\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-28}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-90}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-233}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-235}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-225}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-225}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ z y))))
   (if (<= x -1.05e-6)
     t_0
     (if (<= x -3.2e-28)
       (* z 5.0)
       (if (<= x -2.6e-90)
         (* x y)
         (if (<= x -7e-233)
           (* z 5.0)
           (if (<= x -1.75e-235)
             (* x y)
             (if (<= x 5.2e-225)
               (* z 5.0)
               (if (<= x 5.3e-225)
                 (* x y)
                 (if (<= x 1.9e-37) (* z 5.0) t_0))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z + y);
	double tmp;
	if (x <= -1.05e-6) {
		tmp = t_0;
	} else if (x <= -3.2e-28) {
		tmp = z * 5.0;
	} else if (x <= -2.6e-90) {
		tmp = x * y;
	} else if (x <= -7e-233) {
		tmp = z * 5.0;
	} else if (x <= -1.75e-235) {
		tmp = x * y;
	} else if (x <= 5.2e-225) {
		tmp = z * 5.0;
	} else if (x <= 5.3e-225) {
		tmp = x * y;
	} else if (x <= 1.9e-37) {
		tmp = z * 5.0;
	} 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 <= (-1.05d-6)) then
        tmp = t_0
    else if (x <= (-3.2d-28)) then
        tmp = z * 5.0d0
    else if (x <= (-2.6d-90)) then
        tmp = x * y
    else if (x <= (-7d-233)) then
        tmp = z * 5.0d0
    else if (x <= (-1.75d-235)) then
        tmp = x * y
    else if (x <= 5.2d-225) then
        tmp = z * 5.0d0
    else if (x <= 5.3d-225) then
        tmp = x * y
    else if (x <= 1.9d-37) then
        tmp = z * 5.0d0
    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 <= -1.05e-6) {
		tmp = t_0;
	} else if (x <= -3.2e-28) {
		tmp = z * 5.0;
	} else if (x <= -2.6e-90) {
		tmp = x * y;
	} else if (x <= -7e-233) {
		tmp = z * 5.0;
	} else if (x <= -1.75e-235) {
		tmp = x * y;
	} else if (x <= 5.2e-225) {
		tmp = z * 5.0;
	} else if (x <= 5.3e-225) {
		tmp = x * y;
	} else if (x <= 1.9e-37) {
		tmp = z * 5.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (z + y)
	tmp = 0
	if x <= -1.05e-6:
		tmp = t_0
	elif x <= -3.2e-28:
		tmp = z * 5.0
	elif x <= -2.6e-90:
		tmp = x * y
	elif x <= -7e-233:
		tmp = z * 5.0
	elif x <= -1.75e-235:
		tmp = x * y
	elif x <= 5.2e-225:
		tmp = z * 5.0
	elif x <= 5.3e-225:
		tmp = x * y
	elif x <= 1.9e-37:
		tmp = z * 5.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(z + y))
	tmp = 0.0
	if (x <= -1.05e-6)
		tmp = t_0;
	elseif (x <= -3.2e-28)
		tmp = Float64(z * 5.0);
	elseif (x <= -2.6e-90)
		tmp = Float64(x * y);
	elseif (x <= -7e-233)
		tmp = Float64(z * 5.0);
	elseif (x <= -1.75e-235)
		tmp = Float64(x * y);
	elseif (x <= 5.2e-225)
		tmp = Float64(z * 5.0);
	elseif (x <= 5.3e-225)
		tmp = Float64(x * y);
	elseif (x <= 1.9e-37)
		tmp = Float64(z * 5.0);
	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 <= -1.05e-6)
		tmp = t_0;
	elseif (x <= -3.2e-28)
		tmp = z * 5.0;
	elseif (x <= -2.6e-90)
		tmp = x * y;
	elseif (x <= -7e-233)
		tmp = z * 5.0;
	elseif (x <= -1.75e-235)
		tmp = x * y;
	elseif (x <= 5.2e-225)
		tmp = z * 5.0;
	elseif (x <= 5.3e-225)
		tmp = x * y;
	elseif (x <= 1.9e-37)
		tmp = z * 5.0;
	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, -1.05e-6], t$95$0, If[LessEqual[x, -3.2e-28], N[(z * 5.0), $MachinePrecision], If[LessEqual[x, -2.6e-90], N[(x * y), $MachinePrecision], If[LessEqual[x, -7e-233], N[(z * 5.0), $MachinePrecision], If[LessEqual[x, -1.75e-235], N[(x * y), $MachinePrecision], If[LessEqual[x, 5.2e-225], N[(z * 5.0), $MachinePrecision], If[LessEqual[x, 5.3e-225], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.9e-37], N[(z * 5.0), $MachinePrecision], t$95$0]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0499999999999999e-6 or 1.9000000000000002e-37 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 97.0%

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

      1. +-commutative 97.0%

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

   5. Simplified 97.0%

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

   if -1.0499999999999999e-6 < x < -3.19999999999999982e-28 or -2.6e-90 < x < -6.99999999999999982e-233 or -1.7499999999999999e-235 < x < 5.20000000000000027e-225 or 5.30000000000000005e-225 < x < 1.9000000000000002e-37

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 78.3%

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

   if -3.19999999999999982e-28 < x < -2.6e-90 or -6.99999999999999982e-233 < x < -1.7499999999999999e-235 or 5.20000000000000027e-225 < x < 5.30000000000000005e-225

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 71.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-28}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-90}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-233}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-235}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-225}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-225}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z + y\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-90}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-43}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ z y))))
   (if (<= x -8.5e-6)
     t_0
     (if (<= x -3.5e-14)
       (* z (+ 5.0 x))
       (if (<= x -4.6e-90) (* x y) (if (<= x 7.1e-43) (* z 5.0) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z + y);
	double tmp;
	if (x <= -8.5e-6) {
		tmp = t_0;
	} else if (x <= -3.5e-14) {
		tmp = z * (5.0 + x);
	} else if (x <= -4.6e-90) {
		tmp = x * y;
	} else if (x <= 7.1e-43) {
		tmp = z * 5.0;
	} 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 <= (-8.5d-6)) then
        tmp = t_0
    else if (x <= (-3.5d-14)) then
        tmp = z * (5.0d0 + x)
    else if (x <= (-4.6d-90)) then
        tmp = x * y
    else if (x <= 7.1d-43) then
        tmp = z * 5.0d0
    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 <= -8.5e-6) {
		tmp = t_0;
	} else if (x <= -3.5e-14) {
		tmp = z * (5.0 + x);
	} else if (x <= -4.6e-90) {
		tmp = x * y;
	} else if (x <= 7.1e-43) {
		tmp = z * 5.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (z + y)
	tmp = 0
	if x <= -8.5e-6:
		tmp = t_0
	elif x <= -3.5e-14:
		tmp = z * (5.0 + x)
	elif x <= -4.6e-90:
		tmp = x * y
	elif x <= 7.1e-43:
		tmp = z * 5.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(z + y))
	tmp = 0.0
	if (x <= -8.5e-6)
		tmp = t_0;
	elseif (x <= -3.5e-14)
		tmp = Float64(z * Float64(5.0 + x));
	elseif (x <= -4.6e-90)
		tmp = Float64(x * y);
	elseif (x <= 7.1e-43)
		tmp = Float64(z * 5.0);
	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 <= -8.5e-6)
		tmp = t_0;
	elseif (x <= -3.5e-14)
		tmp = z * (5.0 + x);
	elseif (x <= -4.6e-90)
		tmp = x * y;
	elseif (x <= 7.1e-43)
		tmp = z * 5.0;
	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, -8.5e-6], t$95$0, If[LessEqual[x, -3.5e-14], N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.6e-90], N[(x * y), $MachinePrecision], If[LessEqual[x, 7.1e-43], N[(z * 5.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.4999999999999999e-6 or 7.10000000000000025e-43 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 96.3%

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

      1. +-commutative 96.3%

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

   5. Simplified 96.3%

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

   if -8.4999999999999999e-6 < x < -3.5000000000000002e-14

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

   5. Simplified 100.0%

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

   if -3.5000000000000002e-14 < x < -4.5999999999999996e-90

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 59.7%

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

   if -4.5999999999999996e-90 < x < 7.10000000000000025e-43

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.4%

      \[\leadsto \color{blue}{5 \cdot z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-90}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-43}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-6} \lor \neg \left(x \leq -6.1 \cdot 10^{-37}\right) \land \left(x \leq -4.4 \cdot 10^{-91} \lor \neg \left(x \leq 9.5 \cdot 10^{-29}\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e-6)
         (and (not (<= x -6.1e-37)) (or (<= x -4.4e-91) (not (<= x 9.5e-29)))))
   (* x y)
   (* z 5.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e-6) || (!(x <= -6.1e-37) && ((x <= -4.4e-91) || !(x <= 9.5e-29)))) {
		tmp = x * 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 <= (-1.1d-6)) .or. (.not. (x <= (-6.1d-37))) .and. (x <= (-4.4d-91)) .or. (.not. (x <= 9.5d-29))) then
        tmp = x * 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 <= -1.1e-6) || (!(x <= -6.1e-37) && ((x <= -4.4e-91) || !(x <= 9.5e-29)))) {
		tmp = x * y;
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.1e-6) or (not (x <= -6.1e-37) and ((x <= -4.4e-91) or not (x <= 9.5e-29))):
		tmp = x * y
	else:
		tmp = z * 5.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.1e-6) || (!(x <= -6.1e-37) && ((x <= -4.4e-91) || !(x <= 9.5e-29))))
		tmp = Float64(x * y);
	else
		tmp = Float64(z * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.1e-6) || (~((x <= -6.1e-37)) && ((x <= -4.4e-91) || ~((x <= 9.5e-29)))))
		tmp = x * y;
	else
		tmp = z * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.1e-6], And[N[Not[LessEqual[x, -6.1e-37]], $MachinePrecision], Or[LessEqual[x, -4.4e-91], N[Not[LessEqual[x, 9.5e-29]], $MachinePrecision]]]], N[(x * y), $MachinePrecision], N[(z * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001e-6 or -6.1000000000000003e-37 < x < -4.4000000000000002e-91 or 9.50000000000000023e-29 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 55.9%

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

   if -1.1000000000000001e-6 < x < -6.1000000000000003e-37 or -4.4000000000000002e-91 < x < 9.50000000000000023e-29

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.6%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-6} \lor \neg \left(x \leq -6.1 \cdot 10^{-37}\right) \land \left(x \leq -4.4 \cdot 10^{-91} \lor \neg \left(x \leq 9.5 \cdot 10^{-29}\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 1.26 \cdot 10^{-31}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5 + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.0) (not (<= x 1.26e-31)))
   (* x (+ z y))
   (+ (* z 5.0) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.0) || !(x <= 1.26e-31)) {
		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 <= (-5.0d0)) .or. (.not. (x <= 1.26d-31))) 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 <= -5.0) || !(x <= 1.26e-31)) {
		tmp = x * (z + y);
	} else {
		tmp = (z * 5.0) + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.0) or not (x <= 1.26e-31):
		tmp = x * (z + y)
	else:
		tmp = (z * 5.0) + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.0) || !(x <= 1.26e-31))
		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 <= -5.0) || ~((x <= 1.26e-31)))
		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, -5.0], N[Not[LessEqual[x, 1.26e-31]], $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 < -5 or 1.2599999999999999e-31 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 96.9%

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

      1. +-commutative 96.9%

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

   5. Simplified 96.9%

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

   if -5 < x < 1.2599999999999999e-31

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 89.4%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot x + -1 \cdot \frac{5 \cdot z + x \cdot z}{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 89.4%

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

      2. mul-1-neg 89.4%

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

      3. distribute-lft-out 89.4%

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

      4. distribute-rgt-in 89.4%

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

      5. associate-/l* 89.2%

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

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\left(-y\right) \cdot \left(-1 \cdot \left(x + z \cdot \frac{5 + x}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. clear-num 89.1%

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

      2. un-div-inv 89.2%

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

   7. Applied egg-rr 89.2%

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

   8. Taylor expanded in x around 0 88.3%

      \[\leadsto \left(-y\right) \cdot \left(-1 \cdot \left(x + \frac{z}{\color{blue}{0.2 \cdot y}}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 88.3%

         \[\leadsto \left(-y\right) \cdot \left(-1 \cdot \left(x + \frac{z}{\color{blue}{y \cdot 0.2}}\right)\right) \]

   10. Simplified 88.3%

       \[\leadsto \left(-y\right) \cdot \left(-1 \cdot \left(x + \frac{z}{\color{blue}{y \cdot 0.2}}\right)\right) \]

   11. Taylor expanded in y around 0 99.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 1.26 \cdot 10^{-31}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5 + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto x \cdot \left(z + y\right) + z \cdot 5 \]
4. Add Preprocessing

Alternative 8: 37.3% 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. Add Preprocessing

3. Taylor expanded in x around 0 40.4%

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

4. Final simplification 40.4%

   \[\leadsto z \cdot 5 \]
5. Add Preprocessing

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 2024107 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

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

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