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

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

Alternatives: 13

Speedup: 1.0×

• 27.9% 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 99.9%

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

   1. fma-define 99.9%

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

   2. associate-+l+ 99.9%

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

   3. +-commutative 99.9%

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

   4. count-2 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+16} \lor \neg \left(y \leq -1.3 \cdot 10^{-18} \lor \neg \left(y \leq -1.1 \cdot 10^{-75}\right) \land y \leq 5.8 \cdot 10^{+54}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + 2 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.4e+16)
         (not (or (<= y -1.3e-18) (and (not (<= y -1.1e-75)) (<= y 5.8e+54)))))
   (* 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 <= -2.4e+16) || !((y <= -1.3e-18) || (!(y <= -1.1e-75) && (y <= 5.8e+54)))) {
		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 <= (-2.4d+16)) .or. (.not. (y <= (-1.3d-18)) .or. (.not. (y <= (-1.1d-75))) .and. (y <= 5.8d+54))) 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 <= -2.4e+16) || !((y <= -1.3e-18) || (!(y <= -1.1e-75) && (y <= 5.8e+54)))) {
		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 <= -2.4e+16) or not ((y <= -1.3e-18) or (not (y <= -1.1e-75) and (y <= 5.8e+54))):
		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 <= -2.4e+16) || !((y <= -1.3e-18) || (!(y <= -1.1e-75) && (y <= 5.8e+54))))
		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 <= -2.4e+16) || ~(((y <= -1.3e-18) || (~((y <= -1.1e-75)) && (y <= 5.8e+54)))))
		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, -2.4e+16], N[Not[Or[LessEqual[y, -1.3e-18], And[N[Not[LessEqual[y, -1.1e-75]], $MachinePrecision], LessEqual[y, 5.8e+54]]]], $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 < -2.4e16 or -1.3e-18 < y < -1.10000000000000003e-75 or 5.7999999999999997e54 < 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 78.0%

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

   5. Simplified 78.0%

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

   if -2.4e16 < y < -1.3e-18 or -1.10000000000000003e-75 < y < 5.7999999999999997e54

   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 0 81.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+16} \lor \neg \left(y \leq -1.3 \cdot 10^{-18} \lor \neg \left(y \leq -1.1 \cdot 10^{-75}\right) \land y \leq 5.8 \cdot 10^{+54}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + 2 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(t + 2 \cdot y\right)\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{+45}:\\ \;\;\;\;z \cdot \left(x \cdot 2\right)\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-54} \lor \neg \left(x \leq 2.6 \cdot 10^{-14}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ t (* 2.0 y)))))
   (if (<= x -9.2e+86)
     t_1
     (if (<= x -1.36e+45)
       (* z (* x 2.0))
       (if (or (<= x -5.5e-54) (not (<= x 2.6e-14))) t_1 (* y 5.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t + (2.0 * y));
	double tmp;
	if (x <= -9.2e+86) {
		tmp = t_1;
	} else if (x <= -1.36e+45) {
		tmp = z * (x * 2.0);
	} else if ((x <= -5.5e-54) || !(x <= 2.6e-14)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = x * (t + (2.0d0 * y))
    if (x <= (-9.2d+86)) then
        tmp = t_1
    else if (x <= (-1.36d+45)) then
        tmp = z * (x * 2.0d0)
    else if ((x <= (-5.5d-54)) .or. (.not. (x <= 2.6d-14))) then
        tmp = t_1
    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 t_1 = x * (t + (2.0 * y));
	double tmp;
	if (x <= -9.2e+86) {
		tmp = t_1;
	} else if (x <= -1.36e+45) {
		tmp = z * (x * 2.0);
	} else if ((x <= -5.5e-54) || !(x <= 2.6e-14)) {
		tmp = t_1;
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t + (2.0 * y))
	tmp = 0
	if x <= -9.2e+86:
		tmp = t_1
	elif x <= -1.36e+45:
		tmp = z * (x * 2.0)
	elif (x <= -5.5e-54) or not (x <= 2.6e-14):
		tmp = t_1
	else:
		tmp = y * 5.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t + Float64(2.0 * y)))
	tmp = 0.0
	if (x <= -9.2e+86)
		tmp = t_1;
	elseif (x <= -1.36e+45)
		tmp = Float64(z * Float64(x * 2.0));
	elseif ((x <= -5.5e-54) || !(x <= 2.6e-14))
		tmp = t_1;
	else
		tmp = Float64(y * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t + (2.0 * y));
	tmp = 0.0;
	if (x <= -9.2e+86)
		tmp = t_1;
	elseif (x <= -1.36e+45)
		tmp = z * (x * 2.0);
	elseif ((x <= -5.5e-54) || ~((x <= 2.6e-14)))
		tmp = t_1;
	else
		tmp = y * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.2e+86], t$95$1, If[LessEqual[x, -1.36e+45], N[(z * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -5.5e-54], N[Not[LessEqual[x, 2.6e-14]], $MachinePrecision]], t$95$1, N[(y * 5.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.19999999999999958e86 or -1.36e45 < x < -5.50000000000000046e-54 or 2.59999999999999997e-14 < 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-define 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 99.2%

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

   6. Taylor expanded in z around 0 70.7%

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

   if -9.19999999999999958e86 < x < -1.36e45

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 76.3%

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

   5. Simplified 76.3%

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

   if -5.50000000000000046e-54 < x < 2.59999999999999997e-14

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 64.6%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \left(t + 2 \cdot y\right)\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{+45}:\\ \;\;\;\;z \cdot \left(x \cdot 2\right)\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-54} \lor \neg \left(x \leq 2.6 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \left(t + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(t + 2 \cdot z\right)\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-126}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+245}:\\ \;\;\;\;\left(y + z\right) \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + 2 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ t (* 2.0 z)))))
   (if (<= x -7.4e-57)
     t_1
     (if (<= x 2.3e-126)
       (+ (* y 5.0) (* x t))
       (if (<= x 2.8e+156)
         t_1
         (if (<= x 6.5e+245) (* (+ y z) (* x 2.0)) (* x (+ t (* 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 <= -7.4e-57) {
		tmp = t_1;
	} else if (x <= 2.3e-126) {
		tmp = (y * 5.0) + (x * t);
	} else if (x <= 2.8e+156) {
		tmp = t_1;
	} else if (x <= 6.5e+245) {
		tmp = (y + z) * (x * 2.0);
	} else {
		tmp = x * (t + (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 <= (-7.4d-57)) then
        tmp = t_1
    else if (x <= 2.3d-126) then
        tmp = (y * 5.0d0) + (x * t)
    else if (x <= 2.8d+156) then
        tmp = t_1
    else if (x <= 6.5d+245) then
        tmp = (y + z) * (x * 2.0d0)
    else
        tmp = x * (t + (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 <= -7.4e-57) {
		tmp = t_1;
	} else if (x <= 2.3e-126) {
		tmp = (y * 5.0) + (x * t);
	} else if (x <= 2.8e+156) {
		tmp = t_1;
	} else if (x <= 6.5e+245) {
		tmp = (y + z) * (x * 2.0);
	} else {
		tmp = x * (t + (2.0 * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t + (2.0 * z))
	tmp = 0
	if x <= -7.4e-57:
		tmp = t_1
	elif x <= 2.3e-126:
		tmp = (y * 5.0) + (x * t)
	elif x <= 2.8e+156:
		tmp = t_1
	elif x <= 6.5e+245:
		tmp = (y + z) * (x * 2.0)
	else:
		tmp = x * (t + (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 <= -7.4e-57)
		tmp = t_1;
	elseif (x <= 2.3e-126)
		tmp = Float64(Float64(y * 5.0) + Float64(x * t));
	elseif (x <= 2.8e+156)
		tmp = t_1;
	elseif (x <= 6.5e+245)
		tmp = Float64(Float64(y + z) * Float64(x * 2.0));
	else
		tmp = Float64(x * Float64(t + 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 <= -7.4e-57)
		tmp = t_1;
	elseif (x <= 2.3e-126)
		tmp = (y * 5.0) + (x * t);
	elseif (x <= 2.8e+156)
		tmp = t_1;
	elseif (x <= 6.5e+245)
		tmp = (y + z) * (x * 2.0);
	else
		tmp = x * (t + (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, -7.4e-57], t$95$1, If[LessEqual[x, 2.3e-126], N[(N[(y * 5.0), $MachinePrecision] + N[(x * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e+156], t$95$1, If[LessEqual[x, 6.5e+245], N[(N[(y + z), $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(t + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.4e-57 or 2.30000000000000011e-126 < x < 2.79999999999999988e156

   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 73.5%

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

   if -7.4e-57 < x < 2.30000000000000011e-126

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 99.8%

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

   4. Taylor expanded in z around 0 87.9%

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

   if 2.79999999999999988e156 < x < 6.50000000000000035e245

   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-define 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. Taylor expanded in t around 0 89.8%

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

      1. associate-*r* 89.8%

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

      2. *-commutative 89.8%

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

   8. Simplified 89.8%

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

   if 6.50000000000000035e245 < 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-define 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. Taylor expanded in z around 0 92.2%

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \left(t + 2 \cdot z\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-126}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+156}:\\ \;\;\;\;x \cdot \left(t + 2 \cdot z\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+245}:\\ \;\;\;\;\left(y + z\right) \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + 2 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-53}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-10}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+162} \lor \neg \left(x \leq 2.8 \cdot 10^{+279}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.3e-53)
   (* x t)
   (if (<= x 4.5e-10)
     (* y 5.0)
     (if (or (<= x 5.8e+162) (not (<= x 2.8e+279))) (* x t) (* y (* x 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.3e-53) {
		tmp = x * t;
	} else if (x <= 4.5e-10) {
		tmp = y * 5.0;
	} else if ((x <= 5.8e+162) || !(x <= 2.8e+279)) {
		tmp = x * t;
	} else {
		tmp = y * (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 <= (-2.3d-53)) then
        tmp = x * t
    else if (x <= 4.5d-10) then
        tmp = y * 5.0d0
    else if ((x <= 5.8d+162) .or. (.not. (x <= 2.8d+279))) then
        tmp = x * t
    else
        tmp = y * (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 <= -2.3e-53) {
		tmp = x * t;
	} else if (x <= 4.5e-10) {
		tmp = y * 5.0;
	} else if ((x <= 5.8e+162) || !(x <= 2.8e+279)) {
		tmp = x * t;
	} else {
		tmp = y * (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.3e-53:
		tmp = x * t
	elif x <= 4.5e-10:
		tmp = y * 5.0
	elif (x <= 5.8e+162) or not (x <= 2.8e+279):
		tmp = x * t
	else:
		tmp = y * (x * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.3e-53)
		tmp = Float64(x * t);
	elseif (x <= 4.5e-10)
		tmp = Float64(y * 5.0);
	elseif ((x <= 5.8e+162) || !(x <= 2.8e+279))
		tmp = Float64(x * t);
	else
		tmp = Float64(y * Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.3e-53)
		tmp = x * t;
	elseif (x <= 4.5e-10)
		tmp = y * 5.0;
	elseif ((x <= 5.8e+162) || ~((x <= 2.8e+279)))
		tmp = x * t;
	else
		tmp = y * (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.3e-53], N[(x * t), $MachinePrecision], If[LessEqual[x, 4.5e-10], N[(y * 5.0), $MachinePrecision], If[Or[LessEqual[x, 5.8e+162], N[Not[LessEqual[x, 2.8e+279]], $MachinePrecision]], N[(x * t), $MachinePrecision], N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e-53 or 4.5e-10 < x < 5.80000000000000012e162 or 2.8000000000000002e279 < 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 42.8%

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

   5. Simplified 42.8%

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

   if -2.3000000000000001e-53 < x < 4.5e-10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 64.6%

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

   if 5.80000000000000012e162 < x < 2.8000000000000002e279

   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 inf 42.1%

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

   5. Simplified 42.1%

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

   6. Taylor expanded in x around inf 42.1%

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

   7. Taylor expanded in x around inf 42.1%

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

      1. associate-*r* 42.1%

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

      2. *-commutative 42.1%

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

   9. Simplified 42.1%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-53}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-10}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+162} \lor \neg \left(x \leq 2.8 \cdot 10^{+279}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5400000 \lor \neg \left(x \leq 3.4 \cdot 10^{-34}\right):\\ \;\;\;\;x \cdot \left(t + \left(2 \cdot \left(y + z\right) + 5 \cdot \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + 2 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5400000.0) (not (<= x 3.4e-34)))
   (* x (+ t (+ (* 2.0 (+ y z)) (* 5.0 (/ y x)))))
   (+ (* y 5.0) (* x (+ t (* 2.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5400000.0) || !(x <= 3.4e-34)) {
		tmp = x * (t + ((2.0 * (y + z)) + (5.0 * (y / x))));
	} else {
		tmp = (y * 5.0) + (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 ((x <= (-5400000.0d0)) .or. (.not. (x <= 3.4d-34))) then
        tmp = x * (t + ((2.0d0 * (y + z)) + (5.0d0 * (y / x))))
    else
        tmp = (y * 5.0d0) + (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 ((x <= -5400000.0) || !(x <= 3.4e-34)) {
		tmp = x * (t + ((2.0 * (y + z)) + (5.0 * (y / x))));
	} else {
		tmp = (y * 5.0) + (x * (t + (2.0 * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5400000.0) or not (x <= 3.4e-34):
		tmp = x * (t + ((2.0 * (y + z)) + (5.0 * (y / x))))
	else:
		tmp = (y * 5.0) + (x * (t + (2.0 * z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5400000.0) || !(x <= 3.4e-34))
		tmp = Float64(x * Float64(t + Float64(Float64(2.0 * Float64(y + z)) + Float64(5.0 * Float64(y / x)))));
	else
		tmp = Float64(Float64(y * 5.0) + 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 ((x <= -5400000.0) || ~((x <= 3.4e-34)))
		tmp = x * (t + ((2.0 * (y + z)) + (5.0 * (y / x))));
	else
		tmp = (y * 5.0) + (x * (t + (2.0 * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5400000.0], N[Not[LessEqual[x, 3.4e-34]], $MachinePrecision]], N[(x * N[(t + N[(N[(2.0 * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(x * N[(t + N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4e6 or 3.4000000000000001e-34 < 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-define 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%

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

   if -5.4e6 < x < 3.4000000000000001e-34

   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 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5400000 \lor \neg \left(x \leq 3.4 \cdot 10^{-34}\right):\\ \;\;\;\;x \cdot \left(t + \left(2 \cdot \left(y + z\right) + 5 \cdot \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + 2 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5400000 \lor \neg \left(x \leq 5.3 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \left(y + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + 2 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5400000.0) (not (<= x 5.3e-8)))
   (* x (+ (* 2.0 (+ y z)) t))
   (+ (* y 5.0) (* x (+ t (* 2.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5400000.0) || !(x <= 5.3e-8)) {
		tmp = x * ((2.0 * (y + z)) + t);
	} else {
		tmp = (y * 5.0) + (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 ((x <= (-5400000.0d0)) .or. (.not. (x <= 5.3d-8))) then
        tmp = x * ((2.0d0 * (y + z)) + t)
    else
        tmp = (y * 5.0d0) + (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 ((x <= -5400000.0) || !(x <= 5.3e-8)) {
		tmp = x * ((2.0 * (y + z)) + t);
	} else {
		tmp = (y * 5.0) + (x * (t + (2.0 * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5400000.0) or not (x <= 5.3e-8):
		tmp = x * ((2.0 * (y + z)) + t)
	else:
		tmp = (y * 5.0) + (x * (t + (2.0 * z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5400000.0) || !(x <= 5.3e-8))
		tmp = Float64(x * Float64(Float64(2.0 * Float64(y + z)) + t));
	else
		tmp = Float64(Float64(y * 5.0) + 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 ((x <= -5400000.0) || ~((x <= 5.3e-8)))
		tmp = x * ((2.0 * (y + z)) + t);
	else
		tmp = (y * 5.0) + (x * (t + (2.0 * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5400000.0], N[Not[LessEqual[x, 5.3e-8]], $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[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4e6 or 5.2999999999999998e-8 < 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-define 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%

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

   if -5.4e6 < x < 5.2999999999999998e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5400000 \lor \neg \left(x \leq 5.3 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \left(y + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + 2 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+156}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (* x 2.0))))
   (if (<= x -4.7e-57)
     t_1
     (if (<= x 1.4e-10) (* y 5.0) (if (<= x 2.5e+156) (* x t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x * 2.0);
	double tmp;
	if (x <= -4.7e-57) {
		tmp = t_1;
	} else if (x <= 1.4e-10) {
		tmp = y * 5.0;
	} else if (x <= 2.5e+156) {
		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 = z * (x * 2.0d0)
    if (x <= (-4.7d-57)) then
        tmp = t_1
    else if (x <= 1.4d-10) then
        tmp = y * 5.0d0
    else if (x <= 2.5d+156) 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 = z * (x * 2.0);
	double tmp;
	if (x <= -4.7e-57) {
		tmp = t_1;
	} else if (x <= 1.4e-10) {
		tmp = y * 5.0;
	} else if (x <= 2.5e+156) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (x * 2.0)
	tmp = 0
	if x <= -4.7e-57:
		tmp = t_1
	elif x <= 1.4e-10:
		tmp = y * 5.0
	elif x <= 2.5e+156:
		tmp = x * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(x * 2.0))
	tmp = 0.0
	if (x <= -4.7e-57)
		tmp = t_1;
	elseif (x <= 1.4e-10)
		tmp = Float64(y * 5.0);
	elseif (x <= 2.5e+156)
		tmp = Float64(x * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (x * 2.0);
	tmp = 0.0;
	if (x <= -4.7e-57)
		tmp = t_1;
	elseif (x <= 1.4e-10)
		tmp = y * 5.0;
	elseif (x <= 2.5e+156)
		tmp = x * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e-57], t$95$1, If[LessEqual[x, 1.4e-10], N[(y * 5.0), $MachinePrecision], If[LessEqual[x, 2.5e+156], N[(x * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6999999999999998e-57 or 2.49999999999999996e156 < 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 z around inf 42.0%

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

   5. Simplified 42.0%

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

   if -4.6999999999999998e-57 < x < 1.40000000000000008e-10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 65.1%

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

   if 1.40000000000000008e-10 < x < 2.49999999999999996e156

   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 50.0%

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

   5. Simplified 50.0%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-57}:\\ \;\;\;\;z \cdot \left(x \cdot 2\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+156}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-56} \lor \neg \left(x \leq 1.06 \cdot 10^{-126}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \left(y + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.15e-56) (not (<= x 1.06e-126)))
   (* 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 <= -2.15e-56) || !(x <= 1.06e-126)) {
		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 <= (-2.15d-56)) .or. (.not. (x <= 1.06d-126))) 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 <= -2.15e-56) || !(x <= 1.06e-126)) {
		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 <= -2.15e-56) or not (x <= 1.06e-126):
		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 <= -2.15e-56) || !(x <= 1.06e-126))
		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 <= -2.15e-56) || ~((x <= 1.06e-126)))
		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, -2.15e-56], N[Not[LessEqual[x, 1.06e-126]], $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 < -2.1500000000000001e-56 or 1.0600000000000001e-126 < 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-define 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 94.5%

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

   if -2.1500000000000001e-56 < x < 1.0600000000000001e-126

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 99.8%

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

   4. Taylor expanded in z around 0 87.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-56} \lor \neg \left(x \leq 1.06 \cdot 10^{-126}\right):\\ \;\;\;\;x \cdot \left(2 \cdot \left(y + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-130} \lor \neg \left(x \leq 3.5 \cdot 10^{-130}\right):\\ \;\;\;\;x \cdot \left(t + 2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.1e-130) (not (<= x 3.5e-130)))
   (* x (+ t (* 2.0 z)))
   (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.1e-130) || !(x <= 3.5e-130)) {
		tmp = x * (t + (2.0 * z));
	} 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 <= (-2.1d-130)) .or. (.not. (x <= 3.5d-130))) then
        tmp = x * (t + (2.0d0 * z))
    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 <= -2.1e-130) || !(x <= 3.5e-130)) {
		tmp = x * (t + (2.0 * z));
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.1e-130) or not (x <= 3.5e-130):
		tmp = x * (t + (2.0 * z))
	else:
		tmp = y * 5.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.1e-130) || !(x <= 3.5e-130))
		tmp = Float64(x * Float64(t + Float64(2.0 * z)));
	else
		tmp = Float64(y * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.1e-130) || ~((x <= 3.5e-130)))
		tmp = x * (t + (2.0 * z));
	else
		tmp = y * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.1e-130], N[Not[LessEqual[x, 3.5e-130]], $MachinePrecision]], N[(x * N[(t + N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.10000000000000002e-130 or 3.4999999999999999e-130 < x

   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 0 70.9%

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

   if -2.10000000000000002e-130 < x < 3.4999999999999999e-130

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 75.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-130} \lor \neg \left(x \leq 3.5 \cdot 10^{-130}\right):\\ \;\;\;\;x \cdot \left(t + 2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 99.9%

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

Alternative 12: 46.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-53} \lor \neg \left(x \leq 1.4 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.2e-53) (not (<= x 1.4e-9))) (* x t) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.2e-53) || !(x <= 1.4e-9)) {
		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 <= (-6.2d-53)) .or. (.not. (x <= 1.4d-9))) 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 <= -6.2e-53) || !(x <= 1.4e-9)) {
		tmp = x * t;
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -6.2e-53) or not (x <= 1.4e-9):
		tmp = x * t
	else:
		tmp = y * 5.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6.2e-53) || !(x <= 1.4e-9))
		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 <= -6.2e-53) || ~((x <= 1.4e-9)))
		tmp = x * t;
	else
		tmp = y * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6.2e-53], N[Not[LessEqual[x, 1.4e-9]], $MachinePrecision]], N[(x * t), $MachinePrecision], N[(y * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000031e-53 or 1.39999999999999992e-9 < 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 39.5%

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

   5. Simplified 39.5%

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

   if -6.20000000000000031e-53 < x < 1.39999999999999992e-9

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 64.6%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-53} \lor \neg \left(x \leq 1.4 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.6% 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 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 31.8%

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

4. Final simplification 31.8%

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

Reproduce

herbie shell --seed 2024100 
(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)))