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

Percentage Accurate: 99.9% → 99.9%

Time: 9.9s

Alternatives: 11

Speedup: 1.0×

• 26.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 2 \cdot \left(y + z\right) + t, y \cdot 5\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma x (+ (* 2.0 (+ y z)) t) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return fma(x, ((2.0 * (y + z)) + t), (y * 5.0));
}

Julia

function code(x, y, z, t)
	return fma(x, Float64(Float64(2.0 * Float64(y + z)) + t), Float64(y * 5.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(2.0 * N[(y + z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. fma-def 100.0%

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

   2. associate-+l+ 100.0%

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

   3. +-commutative 100.0%

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

   4. count-2 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot \left(y + z\right)} + t, y \cdot 5\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2 \cdot \left(y + z\right) + t, y \cdot 5\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, 2 \cdot \left(y + z\right) + t, y \cdot 5\right) \]
6. Add Preprocessing

Alternative 2: 48.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(2 \cdot y\right)\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+146}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-57}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-59}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+79} \lor \neg \left(x \leq 1.45 \cdot 10^{+147}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (* 2.0 y))))
   (if (<= x -1.1e+146)
     (* x t)
     (if (<= x -3.8e+67)
       t_1
       (if (<= x -4.2e+34)
         (* x t)
         (if (<= x -9.6e+19)
           t_1
           (if (<= x -2e-57)
             (* 2.0 (* x z))
             (if (<= x 2.4e-59)
               (* y 5.0)
               (if (or (<= x 4.7e+79) (not (<= x 1.45e+147)))
                 (* x t)
                 t_1)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (2.0 * y);
	double tmp;
	if (x <= -1.1e+146) {
		tmp = x * t;
	} else if (x <= -3.8e+67) {
		tmp = t_1;
	} else if (x <= -4.2e+34) {
		tmp = x * t;
	} else if (x <= -9.6e+19) {
		tmp = t_1;
	} else if (x <= -2e-57) {
		tmp = 2.0 * (x * z);
	} else if (x <= 2.4e-59) {
		tmp = y * 5.0;
	} else if ((x <= 4.7e+79) || !(x <= 1.45e+147)) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (2.0d0 * y)
    if (x <= (-1.1d+146)) then
        tmp = x * t
    else if (x <= (-3.8d+67)) then
        tmp = t_1
    else if (x <= (-4.2d+34)) then
        tmp = x * t
    else if (x <= (-9.6d+19)) then
        tmp = t_1
    else if (x <= (-2d-57)) then
        tmp = 2.0d0 * (x * z)
    else if (x <= 2.4d-59) then
        tmp = y * 5.0d0
    else if ((x <= 4.7d+79) .or. (.not. (x <= 1.45d+147))) then
        tmp = x * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (2.0 * y);
	double tmp;
	if (x <= -1.1e+146) {
		tmp = x * t;
	} else if (x <= -3.8e+67) {
		tmp = t_1;
	} else if (x <= -4.2e+34) {
		tmp = x * t;
	} else if (x <= -9.6e+19) {
		tmp = t_1;
	} else if (x <= -2e-57) {
		tmp = 2.0 * (x * z);
	} else if (x <= 2.4e-59) {
		tmp = y * 5.0;
	} else if ((x <= 4.7e+79) || !(x <= 1.45e+147)) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (2.0 * y)
	tmp = 0
	if x <= -1.1e+146:
		tmp = x * t
	elif x <= -3.8e+67:
		tmp = t_1
	elif x <= -4.2e+34:
		tmp = x * t
	elif x <= -9.6e+19:
		tmp = t_1
	elif x <= -2e-57:
		tmp = 2.0 * (x * z)
	elif x <= 2.4e-59:
		tmp = y * 5.0
	elif (x <= 4.7e+79) or not (x <= 1.45e+147):
		tmp = x * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(2.0 * y))
	tmp = 0.0
	if (x <= -1.1e+146)
		tmp = Float64(x * t);
	elseif (x <= -3.8e+67)
		tmp = t_1;
	elseif (x <= -4.2e+34)
		tmp = Float64(x * t);
	elseif (x <= -9.6e+19)
		tmp = t_1;
	elseif (x <= -2e-57)
		tmp = Float64(2.0 * Float64(x * z));
	elseif (x <= 2.4e-59)
		tmp = Float64(y * 5.0);
	elseif ((x <= 4.7e+79) || !(x <= 1.45e+147))
		tmp = Float64(x * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (2.0 * y);
	tmp = 0.0;
	if (x <= -1.1e+146)
		tmp = x * t;
	elseif (x <= -3.8e+67)
		tmp = t_1;
	elseif (x <= -4.2e+34)
		tmp = x * t;
	elseif (x <= -9.6e+19)
		tmp = t_1;
	elseif (x <= -2e-57)
		tmp = 2.0 * (x * z);
	elseif (x <= 2.4e-59)
		tmp = y * 5.0;
	elseif ((x <= 4.7e+79) || ~((x <= 1.45e+147)))
		tmp = x * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.1e+146], N[(x * t), $MachinePrecision], If[LessEqual[x, -3.8e+67], t$95$1, If[LessEqual[x, -4.2e+34], N[(x * t), $MachinePrecision], If[LessEqual[x, -9.6e+19], t$95$1, If[LessEqual[x, -2e-57], N[(2.0 * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-59], N[(y * 5.0), $MachinePrecision], If[Or[LessEqual[x, 4.7e+79], N[Not[LessEqual[x, 1.45e+147]], $MachinePrecision]], N[(x * t), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.0999999999999999e146 or -3.8000000000000002e67 < x < -4.20000000000000035e34 or 2.40000000000000015e-59 < x < 4.70000000000000023e79 or 1.4499999999999999e147 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 55.5%

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

   5. Simplified 55.5%

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

   if -1.0999999999999999e146 < x < -3.8000000000000002e67 or -4.20000000000000035e34 < x < -9.6e19 or 4.70000000000000023e79 < x < 1.4499999999999999e147

   1. Initial program 100.0%

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

      1. fma-def 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot \left(y + z\right)} + t, y \cdot 5\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2 \cdot \left(y + z\right) + t, y \cdot 5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot \left(y + z\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. fma-def 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around inf 65.1%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 65.1%

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

      2. *-commutative 65.1%

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

      3. associate-*r* 65.1%

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

   10. Simplified 65.1%

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

   if -9.6e19 < x < -1.99999999999999991e-57

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf 47.0%

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

   if -1.99999999999999991e-57 < x < 2.40000000000000015e-59

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 69.9%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+146}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \left(2 \cdot y\right)\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(2 \cdot y\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-57}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-59}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+79} \lor \neg \left(x \leq 1.45 \cdot 10^{+147}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(t + 2 \cdot z\right)\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-63}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{+82} \lor \neg \left(x \leq 1.05 \cdot 10^{+144}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ t (* 2.0 z)))))
   (if (<= x -1.55e-57)
     t_1
     (if (<= x 4.3e-63)
       (* y 5.0)
       (if (or (<= x 1.08e+82) (not (<= x 1.05e+144))) t_1 (* x (* 2.0 y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t + (2.0 * z));
	double tmp;
	if (x <= -1.55e-57) {
		tmp = t_1;
	} else if (x <= 4.3e-63) {
		tmp = y * 5.0;
	} else if ((x <= 1.08e+82) || !(x <= 1.05e+144)) {
		tmp = t_1;
	} else {
		tmp = x * (2.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (t + (2.0d0 * z))
    if (x <= (-1.55d-57)) then
        tmp = t_1
    else if (x <= 4.3d-63) then
        tmp = y * 5.0d0
    else if ((x <= 1.08d+82) .or. (.not. (x <= 1.05d+144))) then
        tmp = t_1
    else
        tmp = x * (2.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (t + (2.0 * z));
	double tmp;
	if (x <= -1.55e-57) {
		tmp = t_1;
	} else if (x <= 4.3e-63) {
		tmp = y * 5.0;
	} else if ((x <= 1.08e+82) || !(x <= 1.05e+144)) {
		tmp = t_1;
	} else {
		tmp = x * (2.0 * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t + (2.0 * z))
	tmp = 0
	if x <= -1.55e-57:
		tmp = t_1
	elif x <= 4.3e-63:
		tmp = y * 5.0
	elif (x <= 1.08e+82) or not (x <= 1.05e+144):
		tmp = t_1
	else:
		tmp = x * (2.0 * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t + Float64(2.0 * z)))
	tmp = 0.0
	if (x <= -1.55e-57)
		tmp = t_1;
	elseif (x <= 4.3e-63)
		tmp = Float64(y * 5.0);
	elseif ((x <= 1.08e+82) || !(x <= 1.05e+144))
		tmp = t_1;
	else
		tmp = Float64(x * Float64(2.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t + (2.0 * z));
	tmp = 0.0;
	if (x <= -1.55e-57)
		tmp = t_1;
	elseif (x <= 4.3e-63)
		tmp = y * 5.0;
	elseif ((x <= 1.08e+82) || ~((x <= 1.05e+144)))
		tmp = t_1;
	else
		tmp = x * (2.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t + N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.55e-57], t$95$1, If[LessEqual[x, 4.3e-63], N[(y * 5.0), $MachinePrecision], If[Or[LessEqual[x, 1.08e+82], N[Not[LessEqual[x, 1.05e+144]], $MachinePrecision]], t$95$1, N[(x * N[(2.0 * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.54999999999999988e-57 or 4.2999999999999999e-63 < x < 1.08e82 or 1.04999999999999998e144 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 77.4%

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

   if -1.54999999999999988e-57 < x < 4.2999999999999999e-63

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 70.6%

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

   if 1.08e82 < x < 1.04999999999999998e144

   1. Initial program 100.0%

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

      1. fma-def 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot \left(y + z\right)} + t, y \cdot 5\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2 \cdot \left(y + z\right) + t, y \cdot 5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot \left(y + z\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. fma-def 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around inf 69.6%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-*r* 69.6%

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

   10. Simplified 69.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \left(t + 2 \cdot z\right)\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-63}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{+82} \lor \neg \left(x \leq 1.05 \cdot 10^{+144}\right):\\ \;\;\;\;x \cdot \left(t + 2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+152}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-57} \lor \neg \left(x \leq 1.85 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -7.6e+152)
   (* x t)
   (if (<= x -2.65e+72)
     (* 2.0 (* x z))
     (if (or (<= x -1.65e-57) (not (<= x 1.85e-59))) (* x t) (* y 5.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.6e+152) {
		tmp = x * t;
	} else if (x <= -2.65e+72) {
		tmp = 2.0 * (x * z);
	} else if ((x <= -1.65e-57) || !(x <= 1.85e-59)) {
		tmp = x * t;
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-7.6d+152)) then
        tmp = x * t
    else if (x <= (-2.65d+72)) then
        tmp = 2.0d0 * (x * z)
    else if ((x <= (-1.65d-57)) .or. (.not. (x <= 1.85d-59))) then
        tmp = x * t
    else
        tmp = y * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.6e+152) {
		tmp = x * t;
	} else if (x <= -2.65e+72) {
		tmp = 2.0 * (x * z);
	} else if ((x <= -1.65e-57) || !(x <= 1.85e-59)) {
		tmp = x * t;
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -7.6e+152:
		tmp = x * t
	elif x <= -2.65e+72:
		tmp = 2.0 * (x * z)
	elif (x <= -1.65e-57) or not (x <= 1.85e-59):
		tmp = x * t
	else:
		tmp = y * 5.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -7.6e+152)
		tmp = Float64(x * t);
	elseif (x <= -2.65e+72)
		tmp = Float64(2.0 * Float64(x * z));
	elseif ((x <= -1.65e-57) || !(x <= 1.85e-59))
		tmp = Float64(x * t);
	else
		tmp = Float64(y * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -7.6e+152)
		tmp = x * t;
	elseif (x <= -2.65e+72)
		tmp = 2.0 * (x * z);
	elseif ((x <= -1.65e-57) || ~((x <= 1.85e-59)))
		tmp = x * t;
	else
		tmp = y * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -7.6e+152], N[(x * t), $MachinePrecision], If[LessEqual[x, -2.65e+72], N[(2.0 * N[(x * z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -1.65e-57], N[Not[LessEqual[x, 1.85e-59]], $MachinePrecision]], N[(x * t), $MachinePrecision], N[(y * 5.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.6000000000000001e152 or -2.6500000000000001e72 < x < -1.6499999999999999e-57 or 1.85e-59 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 49.8%

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

   5. Simplified 49.8%

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

   if -7.6000000000000001e152 < x < -2.6500000000000001e72

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 48.0%

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

   if -1.6499999999999999e-57 < x < 1.85e-59

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 69.9%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+152}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-57} \lor \neg \left(x \leq 1.85 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-47} \lor \neg \left(x \leq 8.6 \cdot 10^{-60}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \left(y + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + \left(y + y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.1e-47) (not (<= x 8.6e-60)))
   (* x (+ (* 2.0 (+ y z)) t))
   (+ (* y 5.0) (* x (+ t (+ y y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.1e-47) || !(x <= 8.6e-60)) {
		tmp = x * ((2.0 * (y + z)) + t);
	} else {
		tmp = (y * 5.0) + (x * (t + (y + y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-4.1d-47)) .or. (.not. (x <= 8.6d-60))) then
        tmp = x * ((2.0d0 * (y + z)) + t)
    else
        tmp = (y * 5.0d0) + (x * (t + (y + y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.1e-47) || !(x <= 8.6e-60)) {
		tmp = x * ((2.0 * (y + z)) + t);
	} else {
		tmp = (y * 5.0) + (x * (t + (y + y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.1e-47) or not (x <= 8.6e-60):
		tmp = x * ((2.0 * (y + z)) + t)
	else:
		tmp = (y * 5.0) + (x * (t + (y + y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.1e-47) || !(x <= 8.6e-60))
		tmp = Float64(x * Float64(Float64(2.0 * Float64(y + z)) + t));
	else
		tmp = Float64(Float64(y * 5.0) + Float64(x * Float64(t + Float64(y + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.1e-47) || ~((x <= 8.6e-60)))
		tmp = x * ((2.0 * (y + z)) + t);
	else
		tmp = (y * 5.0) + (x * (t + (y + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.1e-47], N[Not[LessEqual[x, 8.6e-60]], $MachinePrecision]], N[(x * N[(N[(2.0 * N[(y + z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(x * N[(t + N[(y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.10000000000000002e-47 or 8.6000000000000001e-60 < x

   1. Initial program 100.0%

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

      1. fma-def 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot \left(y + z\right)} + t, y \cdot 5\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2 \cdot \left(y + z\right) + t, y \cdot 5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.8%

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

   if -4.10000000000000002e-47 < x < 8.6000000000000001e-60

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 82.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-47} \lor \neg \left(x \leq 8.6 \cdot 10^{-60}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \left(y + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + \left(y + y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-57} \lor \neg \left(x \leq 3.3 \cdot 10^{-62}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \left(y + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + \left(y + z\right) \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.65e-57) (not (<= x 3.3e-62)))
   (* x (+ (* 2.0 (+ y z)) t))
   (+ (* y 5.0) (* (+ y z) (* x 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.65e-57) || !(x <= 3.3e-62)) {
		tmp = x * ((2.0 * (y + z)) + t);
	} else {
		tmp = (y * 5.0) + ((y + z) * (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.65d-57)) .or. (.not. (x <= 3.3d-62))) then
        tmp = x * ((2.0d0 * (y + z)) + t)
    else
        tmp = (y * 5.0d0) + ((y + z) * (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.65e-57) || !(x <= 3.3e-62)) {
		tmp = x * ((2.0 * (y + z)) + t);
	} else {
		tmp = (y * 5.0) + ((y + z) * (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.65e-57) or not (x <= 3.3e-62):
		tmp = x * ((2.0 * (y + z)) + t)
	else:
		tmp = (y * 5.0) + ((y + z) * (x * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.65e-57) || !(x <= 3.3e-62))
		tmp = Float64(x * Float64(Float64(2.0 * Float64(y + z)) + t));
	else
		tmp = Float64(Float64(y * 5.0) + Float64(Float64(y + z) * Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.65e-57) || ~((x <= 3.3e-62)))
		tmp = x * ((2.0 * (y + z)) + t);
	else
		tmp = (y * 5.0) + ((y + z) * (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.65e-57], N[Not[LessEqual[x, 3.3e-62]], $MachinePrecision]], N[(x * N[(N[(2.0 * N[(y + z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(N[(y + z), $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6499999999999999e-57 or 3.30000000000000004e-62 < x

   1. Initial program 100.0%

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

      1. fma-def 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot \left(y + z\right)} + t, y \cdot 5\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2 \cdot \left(y + z\right) + t, y \cdot 5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.8%

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

   if -1.6499999999999999e-57 < x < 3.30000000000000004e-62

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0 86.6%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot y + 2 \cdot z\right)} + y \cdot 5 \]

   5. Simplified 86.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-57} \lor \neg \left(x \leq 3.3 \cdot 10^{-62}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \left(y + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + \left(y + z\right) \cdot \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-47} \lor \neg \left(x \leq 1.4 \cdot 10^{-62}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \left(y + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.5e-47) (not (<= x 1.4e-62)))
   (* x (+ (* 2.0 (+ y z)) t))
   (+ (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e-47) || !(x <= 1.4e-62)) {
		tmp = x * ((2.0 * (y + z)) + t);
	} else {
		tmp = (y * 5.0) + (x * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-4.5d-47)) .or. (.not. (x <= 1.4d-62))) then
        tmp = x * ((2.0d0 * (y + z)) + t)
    else
        tmp = (y * 5.0d0) + (x * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e-47) || !(x <= 1.4e-62)) {
		tmp = x * ((2.0 * (y + z)) + t);
	} else {
		tmp = (y * 5.0) + (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.5e-47) or not (x <= 1.4e-62):
		tmp = x * ((2.0 * (y + z)) + t)
	else:
		tmp = (y * 5.0) + (x * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.5e-47) || !(x <= 1.4e-62))
		tmp = Float64(x * Float64(Float64(2.0 * Float64(y + z)) + t));
	else
		tmp = Float64(Float64(y * 5.0) + Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.5e-47) || ~((x <= 1.4e-62)))
		tmp = x * ((2.0 * (y + z)) + t);
	else
		tmp = (y * 5.0) + (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.5e-47], N[Not[LessEqual[x, 1.4e-62]], $MachinePrecision]], N[(x * N[(N[(2.0 * N[(y + z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(x * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5e-47 or 1.40000000000000001e-62 < x

   1. Initial program 100.0%

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

      1. fma-def 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot \left(y + z\right)} + t, y \cdot 5\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2 \cdot \left(y + z\right) + t, y \cdot 5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.8%

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

   if -4.5e-47 < x < 1.40000000000000001e-62

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. count-2 99.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \left(y + z\right)}, x, t \cdot x\right) + y \cdot 5 \]

      6. *-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y + z\right) \cdot 2}, x, t \cdot x\right) + y \cdot 5 \]

      7. *-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(\left(y + z\right) \cdot 2, x, \color{blue}{x \cdot t}\right) + y \cdot 5 \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + z\right) \cdot 2, x, x \cdot t\right)} + y \cdot 5 \]

   5. Taylor expanded in t around inf 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-47} \lor \neg \left(x \leq 1.4 \cdot 10^{-62}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \left(y + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+32} \lor \neg \left(y \leq 9.8 \cdot 10^{+26}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + 2 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -7.6e+32) (not (<= y 9.8e+26)))
   (* y (+ 5.0 (* x 2.0)))
   (* x (+ t (* 2.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -7.6e+32) || !(y <= 9.8e+26)) {
		tmp = y * (5.0 + (x * 2.0));
	} else {
		tmp = x * (t + (2.0 * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-7.6d+32)) .or. (.not. (y <= 9.8d+26))) then
        tmp = y * (5.0d0 + (x * 2.0d0))
    else
        tmp = x * (t + (2.0d0 * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -7.6e+32) || !(y <= 9.8e+26)) {
		tmp = y * (5.0 + (x * 2.0));
	} else {
		tmp = x * (t + (2.0 * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -7.6e+32) or not (y <= 9.8e+26):
		tmp = y * (5.0 + (x * 2.0))
	else:
		tmp = x * (t + (2.0 * z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -7.6e+32) || !(y <= 9.8e+26))
		tmp = Float64(y * Float64(5.0 + Float64(x * 2.0)));
	else
		tmp = Float64(x * Float64(t + Float64(2.0 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -7.6e+32) || ~((y <= 9.8e+26)))
		tmp = y * (5.0 + (x * 2.0));
	else
		tmp = x * (t + (2.0 * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -7.6e+32], N[Not[LessEqual[y, 9.8e+26]], $MachinePrecision]], N[(y * N[(5.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t + N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.6000000000000006e32 or 9.79999999999999947e26 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 82.7%

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]

   5. Simplified 82.7%

      \[\leadsto \color{blue}{y \cdot \left(5 + x \cdot 2\right)} \]

   if -7.6000000000000006e32 < y < 9.79999999999999947e26

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 83.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+32} \lor \neg \left(y \leq 9.8 \cdot 10^{+26}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + 2 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 10: 47.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-57} \lor \neg \left(x \leq 1.45 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.45e-57) (not (<= x 1.45e-59))) (* x t) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.45e-57) || !(x <= 1.45e-59)) {
		tmp = x * t;
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.45d-57)) .or. (.not. (x <= 1.45d-59))) then
        tmp = x * t
    else
        tmp = y * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.45e-57) || !(x <= 1.45e-59)) {
		tmp = x * t;
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.45e-57) or not (x <= 1.45e-59):
		tmp = x * t
	else:
		tmp = y * 5.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.45e-57) || !(x <= 1.45e-59))
		tmp = Float64(x * t);
	else
		tmp = Float64(y * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.45e-57) || ~((x <= 1.45e-59)))
		tmp = x * t;
	else
		tmp = y * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.45e-57], N[Not[LessEqual[x, 1.45e-59]], $MachinePrecision]], N[(x * t), $MachinePrecision], N[(y * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.44999999999999994e-57 or 1.45000000000000008e-59 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 46.6%

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

   5. Simplified 46.6%

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

   if -2.44999999999999994e-57 < x < 1.45000000000000008e-59

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 69.9%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-57} \lor \neg \left(x \leq 1.45 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 27.2%

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

4. Final simplification 27.2%

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

Reproduce

herbie shell --seed 2024018 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))