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

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 15

Speedup: 1.0×

• 27.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. flip-+ 61.0%

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

   4. associate-*r/ 53.8%

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

   5. fma-neg 55.8%

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

   6. associate-+l+ 55.9%

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

   7. +-commutative 55.9%

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

   8. count-2 55.9%

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

   9. associate-+l+ 55.8%

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

   10. +-commutative 55.8%

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

   11. count-2 55.8%

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

   12. fma-neg 53.8%

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

   13. associate-+l+ 53.8%

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

   14. +-commutative 53.8%

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

   15. count-2 53.8%

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

4. Applied egg-rr 99.6%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. fma-define 99.5%

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

   2. associate-+l+ 99.5%

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

   3. +-commutative 99.5%

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

   4. count-2 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+81} \lor \neg \left(y \leq 1.4 \cdot 10^{+123}\right):\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + \left(\left(y + z\right) \cdot 2 + 5 \cdot \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.25e+81) (not (<= y 1.4e+123)))
   (+ (* y 5.0) (* 2.0 (* x (+ y z))))
   (* x (+ t (+ (* (+ y z) 2.0) (* 5.0 (/ y x)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.25e+81) || !(y <= 1.4e+123)) {
		tmp = (y * 5.0) + (2.0 * (x * (y + z)));
	} else {
		tmp = x * (t + (((y + z) * 2.0) + (5.0 * (y / x))));
	}
	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 <= (-1.25d+81)) .or. (.not. (y <= 1.4d+123))) then
        tmp = (y * 5.0d0) + (2.0d0 * (x * (y + z)))
    else
        tmp = x * (t + (((y + z) * 2.0d0) + (5.0d0 * (y / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.25e+81) || !(y <= 1.4e+123)) {
		tmp = (y * 5.0) + (2.0 * (x * (y + z)));
	} else {
		tmp = x * (t + (((y + z) * 2.0) + (5.0 * (y / x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.25e+81) or not (y <= 1.4e+123):
		tmp = (y * 5.0) + (2.0 * (x * (y + z)))
	else:
		tmp = x * (t + (((y + z) * 2.0) + (5.0 * (y / x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.25e+81) || !(y <= 1.4e+123))
		tmp = Float64(Float64(y * 5.0) + Float64(2.0 * Float64(x * Float64(y + z))));
	else
		tmp = Float64(x * Float64(t + Float64(Float64(Float64(y + z) * 2.0) + Float64(5.0 * Float64(y / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.25e+81) || ~((y <= 1.4e+123)))
		tmp = (y * 5.0) + (2.0 * (x * (y + z)));
	else
		tmp = x * (t + (((y + z) * 2.0) + (5.0 * (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.25e+81], N[Not[LessEqual[y, 1.4e+123]], $MachinePrecision]], N[(N[(y * 5.0), $MachinePrecision] + N[(2.0 * N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t + N[(N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision] + N[(5.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e81 or 1.40000000000000006e123 < y

   1. Initial program 98.8%

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

      1. fma-define 98.8%

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

      2. associate-+l+ 98.8%

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

      3. +-commutative 98.8%

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

      4. count-2 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t around 0 95.1%

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

   if -1.25e81 < y < 1.40000000000000006e123

   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 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 97.5%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+81} \lor \neg \left(y \leq 1.4 \cdot 10^{+123}\right):\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + \left(\left(y + z\right) \cdot 2 + 5 \cdot \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+241}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+61} \lor \neg \left(y \leq 5.8 \cdot 10^{+26}\right):\\ \;\;\;\;x \cdot \left(\left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.5e+241)
   (* y 5.0)
   (if (or (<= y -1.15e+61) (not (<= y 5.8e+26)))
     (* x (* (+ y z) 2.0))
     (* x (+ t (* z 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.5e+241) {
		tmp = y * 5.0;
	} else if ((y <= -1.15e+61) || !(y <= 5.8e+26)) {
		tmp = x * ((y + z) * 2.0);
	} else {
		tmp = x * (t + (z * 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 (y <= (-4.5d+241)) then
        tmp = y * 5.0d0
    else if ((y <= (-1.15d+61)) .or. (.not. (y <= 5.8d+26))) then
        tmp = x * ((y + z) * 2.0d0)
    else
        tmp = x * (t + (z * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.5e+241) {
		tmp = y * 5.0;
	} else if ((y <= -1.15e+61) || !(y <= 5.8e+26)) {
		tmp = x * ((y + z) * 2.0);
	} else {
		tmp = x * (t + (z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -4.5e+241:
		tmp = y * 5.0
	elif (y <= -1.15e+61) or not (y <= 5.8e+26):
		tmp = x * ((y + z) * 2.0)
	else:
		tmp = x * (t + (z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.5e+241)
		tmp = Float64(y * 5.0);
	elseif ((y <= -1.15e+61) || !(y <= 5.8e+26))
		tmp = Float64(x * Float64(Float64(y + z) * 2.0));
	else
		tmp = Float64(x * Float64(t + Float64(z * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -4.5e+241)
		tmp = y * 5.0;
	elseif ((y <= -1.15e+61) || ~((y <= 5.8e+26)))
		tmp = x * ((y + z) * 2.0);
	else
		tmp = x * (t + (z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.5e+241], N[(y * 5.0), $MachinePrecision], If[Or[LessEqual[y, -1.15e+61], N[Not[LessEqual[y, 5.8e+26]], $MachinePrecision]], N[(x * N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(t + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.49999999999999993e241

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. fma-define 94.4%

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

      3. flip-+ 18.1%

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

      4. associate-*r/ 18.1%

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

      5. fma-neg 18.9%

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

      6. associate-+l+ 18.9%

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

      7. +-commutative 18.9%

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

      8. count-2 18.9%

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

      9. associate-+l+ 18.9%

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

      10. +-commutative 18.9%

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

      11. count-2 18.9%

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

      12. fma-neg 18.1%

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

      13. associate-+l+ 18.1%

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

      14. +-commutative 18.1%

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

      15. count-2 18.1%

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

   4. Applied egg-rr 94.4%

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

   5. Taylor expanded in x around 0 83.6%

      \[\leadsto \color{blue}{5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -4.49999999999999993e241 < y < -1.15e61 or 5.8e26 < 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. 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. Taylor expanded in x around inf 79.8%

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

   6. Taylor expanded in t around 0 77.8%

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

   7. Taylor expanded in x around inf 59.6%

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

      1. +-commutative 59.6%

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

   9. Simplified 59.6%

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

   10. Taylor expanded in t around 0 52.9%

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

   if -1.15e61 < y < 5.8e26

   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. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. flip-+ 69.1%

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

      4. associate-*r/ 63.5%

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

      5. fma-neg 65.8%

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

      6. associate-+l+ 65.8%

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

      7. +-commutative 65.8%

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

      8. count-2 65.8%

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

      9. associate-+l+ 65.8%

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

      10. +-commutative 65.8%

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

      11. count-2 65.8%

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

      12. fma-neg 63.5%

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

      13. associate-+l+ 63.5%

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

      14. +-commutative 63.5%

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

      15. count-2 63.5%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 80.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+241}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+61} \lor \neg \left(y \leq 5.8 \cdot 10^{+26}\right):\\ \;\;\;\;x \cdot \left(\left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+48} \lor \neg \left(y \leq 1.2 \cdot 10^{-22}\right):\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.3e+48) (not (<= y 1.2e-22)))
   (+ (* y 5.0) (* 2.0 (* x (+ y z))))
   (* x (+ t (* (+ y z) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.3e+48) || !(y <= 1.2e-22)) {
		tmp = (y * 5.0) + (2.0 * (x * (y + z)));
	} else {
		tmp = x * (t + ((y + z) * 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 ((y <= (-5.3d+48)) .or. (.not. (y <= 1.2d-22))) then
        tmp = (y * 5.0d0) + (2.0d0 * (x * (y + z)))
    else
        tmp = x * (t + ((y + z) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.3e+48) || !(y <= 1.2e-22)) {
		tmp = (y * 5.0) + (2.0 * (x * (y + z)));
	} else {
		tmp = x * (t + ((y + z) * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -5.3e+48) or not (y <= 1.2e-22):
		tmp = (y * 5.0) + (2.0 * (x * (y + z)))
	else:
		tmp = x * (t + ((y + z) * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.3e+48) || !(y <= 1.2e-22))
		tmp = Float64(Float64(y * 5.0) + Float64(2.0 * Float64(x * Float64(y + z))));
	else
		tmp = Float64(x * Float64(t + Float64(Float64(y + z) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -5.3e+48) || ~((y <= 1.2e-22)))
		tmp = (y * 5.0) + (2.0 * (x * (y + z)));
	else
		tmp = x * (t + ((y + z) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.3e+48], N[Not[LessEqual[y, 1.2e-22]], $MachinePrecision]], N[(N[(y * 5.0), $MachinePrecision] + N[(2.0 * N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t + N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.3e48 or 1.20000000000000001e-22 < y

   1. Initial program 99.1%

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

      1. fma-define 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. count-2 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t around 0 92.2%

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

   if -5.3e48 < y < 1.20000000000000001e-22

   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 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 87.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+48} \lor \neg \left(y \leq 1.2 \cdot 10^{-22}\right):\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+111}:\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+22}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot \left(y + z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4e+111)
   (+ (* y 5.0) (* 2.0 (* x z)))
   (if (<= z 4e+22)
     (+ (* y 5.0) (* x (+ t (* y 2.0))))
     (+ (* y 5.0) (* 2.0 (* x (+ y z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4e+111) {
		tmp = (y * 5.0) + (2.0 * (x * z));
	} else if (z <= 4e+22) {
		tmp = (y * 5.0) + (x * (t + (y * 2.0)));
	} else {
		tmp = (y * 5.0) + (2.0 * (x * (y + 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 (z <= (-4d+111)) then
        tmp = (y * 5.0d0) + (2.0d0 * (x * z))
    else if (z <= 4d+22) then
        tmp = (y * 5.0d0) + (x * (t + (y * 2.0d0)))
    else
        tmp = (y * 5.0d0) + (2.0d0 * (x * (y + z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4e+111) {
		tmp = (y * 5.0) + (2.0 * (x * z));
	} else if (z <= 4e+22) {
		tmp = (y * 5.0) + (x * (t + (y * 2.0)));
	} else {
		tmp = (y * 5.0) + (2.0 * (x * (y + z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4e+111:
		tmp = (y * 5.0) + (2.0 * (x * z))
	elif z <= 4e+22:
		tmp = (y * 5.0) + (x * (t + (y * 2.0)))
	else:
		tmp = (y * 5.0) + (2.0 * (x * (y + z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4e+111)
		tmp = Float64(Float64(y * 5.0) + Float64(2.0 * Float64(x * z)));
	elseif (z <= 4e+22)
		tmp = Float64(Float64(y * 5.0) + Float64(x * Float64(t + Float64(y * 2.0))));
	else
		tmp = Float64(Float64(y * 5.0) + Float64(2.0 * Float64(x * Float64(y + z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4e+111)
		tmp = (y * 5.0) + (2.0 * (x * z));
	elseif (z <= 4e+22)
		tmp = (y * 5.0) + (x * (t + (y * 2.0)));
	else
		tmp = (y * 5.0) + (2.0 * (x * (y + z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4e+111], N[(N[(y * 5.0), $MachinePrecision] + N[(2.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+22], N[(N[(y * 5.0), $MachinePrecision] + N[(x * N[(t + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(2.0 * N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.99999999999999983e111

   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 90.0%

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

   if -3.99999999999999983e111 < z < 4e22

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf 93.5%

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

   if 4e22 < z

   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. Taylor expanded in t around 0 91.3%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+111}:\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+22}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot \left(y + z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-14} \lor \neg \left(x \leq 2.65 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8.8e-14) (not (<= x 2.65e-39)))
   (* x (+ t (* (+ y z) 2.0)))
   (+ (* y 5.0) (* 2.0 (* x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.8e-14) || !(x <= 2.65e-39)) {
		tmp = x * (t + ((y + z) * 2.0));
	} else {
		tmp = (y * 5.0) + (2.0 * (x * 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 <= (-8.8d-14)) .or. (.not. (x <= 2.65d-39))) then
        tmp = x * (t + ((y + z) * 2.0d0))
    else
        tmp = (y * 5.0d0) + (2.0d0 * (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.8e-14) || !(x <= 2.65e-39)) {
		tmp = x * (t + ((y + z) * 2.0));
	} else {
		tmp = (y * 5.0) + (2.0 * (x * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -8.8e-14) or not (x <= 2.65e-39):
		tmp = x * (t + ((y + z) * 2.0))
	else:
		tmp = (y * 5.0) + (2.0 * (x * z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -8.8e-14) || !(x <= 2.65e-39))
		tmp = Float64(x * Float64(t + Float64(Float64(y + z) * 2.0)));
	else
		tmp = Float64(Float64(y * 5.0) + Float64(2.0 * Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -8.8e-14) || ~((x <= 2.65e-39)))
		tmp = x * (t + ((y + z) * 2.0));
	else
		tmp = (y * 5.0) + (2.0 * (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -8.8e-14], N[Not[LessEqual[x, 2.65e-39]], $MachinePrecision]], N[(x * N[(t + N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(2.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8000000000000004e-14 or 2.65000000000000002e-39 < 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. 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 96.9%

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

   if -8.8000000000000004e-14 < x < 2.65000000000000002e-39

   1. Initial program 99.1%

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

   3. Taylor expanded in z around inf 81.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-14} \lor \neg \left(x \leq 2.65 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0011 \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.0011) (not (<= x 2.5e-39)))
   (* x (+ t (* (+ y z) 2.0)))
   (+ (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.0011) || !(x <= 2.5e-39)) {
		tmp = x * (t + ((y + z) * 2.0));
	} 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 <= (-0.0011d0)) .or. (.not. (x <= 2.5d-39))) then
        tmp = x * (t + ((y + z) * 2.0d0))
    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 <= -0.0011) || !(x <= 2.5e-39)) {
		tmp = x * (t + ((y + z) * 2.0));
	} else {
		tmp = (y * 5.0) + (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -0.0011) or not (x <= 2.5e-39):
		tmp = x * (t + ((y + z) * 2.0))
	else:
		tmp = (y * 5.0) + (x * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.0011) || !(x <= 2.5e-39))
		tmp = Float64(x * Float64(t + Float64(Float64(y + z) * 2.0)));
	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 <= -0.0011) || ~((x <= 2.5e-39)))
		tmp = x * (t + ((y + z) * 2.0));
	else
		tmp = (y * 5.0) + (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.0011], N[Not[LessEqual[x, 2.5e-39]], $MachinePrecision]], N[(x * N[(t + N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(x * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00110000000000000007 or 2.4999999999999999e-39 < 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. 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 96.8%

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

   if -0.00110000000000000007 < x < 2.4999999999999999e-39

   1. Initial program 99.1%

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

   3. Taylor expanded in t around inf 78.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0011 \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+46} \lor \neg \left(y \leq 8.2 \cdot 10^{-23}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.15e+46) (not (<= y 8.2e-23)))
   (* y (+ 5.0 (* x 2.0)))
   (* x (+ t (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.15e+46) || !(y <= 8.2e-23)) {
		tmp = y * (5.0 + (x * 2.0));
	} else {
		tmp = x * (t + (z * 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 ((y <= (-1.15d+46)) .or. (.not. (y <= 8.2d-23))) then
        tmp = y * (5.0d0 + (x * 2.0d0))
    else
        tmp = x * (t + (z * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.15e+46) || !(y <= 8.2e-23)) {
		tmp = y * (5.0 + (x * 2.0));
	} else {
		tmp = x * (t + (z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.15e+46) or not (y <= 8.2e-23):
		tmp = y * (5.0 + (x * 2.0))
	else:
		tmp = x * (t + (z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.15e+46) || !(y <= 8.2e-23))
		tmp = Float64(y * Float64(5.0 + Float64(x * 2.0)));
	else
		tmp = Float64(x * Float64(t + Float64(z * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.15e+46) || ~((y <= 8.2e-23)))
		tmp = y * (5.0 + (x * 2.0));
	else
		tmp = x * (t + (z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.15e+46], N[Not[LessEqual[y, 8.2e-23]], $MachinePrecision]], N[(y * N[(5.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15e46 or 8.20000000000000059e-23 < y

   1. Initial program 99.1%

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

      1. fma-define 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. count-2 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around inf 81.7%

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

   if -1.15e46 < y < 8.20000000000000059e-23

   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. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. flip-+ 67.4%

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

      4. associate-*r/ 61.2%

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

      5. fma-neg 63.8%

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

      6. associate-+l+ 63.8%

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

      7. +-commutative 63.8%

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

      8. count-2 63.8%

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

      9. associate-+l+ 63.8%

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

      10. +-commutative 63.8%

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

      11. count-2 63.8%

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

      12. fma-neg 61.2%

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

      13. associate-+l+ 61.2%

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

      14. +-commutative 61.2%

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

      15. count-2 61.2%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 85.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+46} \lor \neg \left(y \leq 8.2 \cdot 10^{-23}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-14} \lor \neg \left(x \leq 2.65 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot \left(\left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7.2e-14) (not (<= x 2.65e-39)))
   (* x (* (+ y z) 2.0))
   (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7.2e-14) || !(x <= 2.65e-39)) {
		tmp = x * ((y + z) * 2.0);
	} 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 <= (-7.2d-14)) .or. (.not. (x <= 2.65d-39))) then
        tmp = x * ((y + z) * 2.0d0)
    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 <= -7.2e-14) || !(x <= 2.65e-39)) {
		tmp = x * ((y + z) * 2.0);
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -7.2e-14) or not (x <= 2.65e-39):
		tmp = x * ((y + z) * 2.0)
	else:
		tmp = y * 5.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -7.2e-14) || !(x <= 2.65e-39))
		tmp = Float64(x * Float64(Float64(y + z) * 2.0));
	else
		tmp = Float64(y * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -7.2e-14) || ~((x <= 2.65e-39)))
		tmp = x * ((y + z) * 2.0);
	else
		tmp = y * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -7.2e-14], N[Not[LessEqual[x, 2.65e-39]], $MachinePrecision]], N[(x * N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(y * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.1999999999999996e-14 or 2.65000000000000002e-39 < 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. 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%

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

   6. Taylor expanded in t around 0 95.3%

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

   7. Taylor expanded in x around inf 96.9%

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

      1. +-commutative 96.9%

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

   9. Simplified 96.9%

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

   10. Taylor expanded in t around 0 66.8%

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

   if -7.1999999999999996e-14 < x < 2.65000000000000002e-39

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-define 99.2%

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

      3. flip-+ 46.9%

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

      4. associate-*r/ 46.1%

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

      5. fma-neg 46.3%

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

      6. associate-+l+ 46.3%

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

      7. +-commutative 46.3%

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

      8. count-2 46.3%

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

      9. associate-+l+ 46.3%

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

      10. +-commutative 46.3%

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

      11. count-2 46.3%

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

      12. fma-neg 46.1%

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

      13. associate-+l+ 46.1%

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

      14. +-commutative 46.1%

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

      15. count-2 46.1%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around 0 61.0%

      \[\leadsto \color{blue}{5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 61.0%

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

   7. Simplified 61.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-14} \lor \neg \left(x \leq 2.65 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot \left(\left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-11}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4e-11) (* x t) (if (<= x 2.5) (* y 5.0) (* y (* x 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4e-11) {
		tmp = x * t;
	} else if (x <= 2.5) {
		tmp = y * 5.0;
	} 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 <= (-4d-11)) then
        tmp = x * t
    else if (x <= 2.5d0) then
        tmp = y * 5.0d0
    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 <= -4e-11) {
		tmp = x * t;
	} else if (x <= 2.5) {
		tmp = y * 5.0;
	} else {
		tmp = y * (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4e-11:
		tmp = x * t
	elif x <= 2.5:
		tmp = y * 5.0
	else:
		tmp = y * (x * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4e-11)
		tmp = Float64(x * t);
	elseif (x <= 2.5)
		tmp = Float64(y * 5.0);
	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 <= -4e-11)
		tmp = x * t;
	elseif (x <= 2.5)
		tmp = y * 5.0;
	else
		tmp = y * (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4e-11], N[(x * t), $MachinePrecision], If[LessEqual[x, 2.5], N[(y * 5.0), $MachinePrecision], N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999976e-11

   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. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. flip-+ 69.8%

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

      4. associate-*r/ 56.0%

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

      5. fma-neg 60.5%

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

      6. associate-+l+ 60.5%

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

      7. +-commutative 60.5%

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

      8. count-2 60.5%

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

      9. associate-+l+ 60.5%

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

      10. +-commutative 60.5%

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

      11. count-2 60.5%

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

      12. fma-neg 56.0%

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

      13. associate-+l+ 56.0%

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

      14. +-commutative 56.0%

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

      15. count-2 56.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around inf 39.6%

      \[\leadsto \color{blue}{t \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 39.6%

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

   7. Simplified 39.6%

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

   if -3.99999999999999976e-11 < x < 2.5

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-define 99.2%

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

      3. flip-+ 47.3%

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

      4. associate-*r/ 46.5%

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

      5. fma-neg 46.7%

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

      6. associate-+l+ 46.7%

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

      7. +-commutative 46.7%

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

      8. count-2 46.7%

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

      9. associate-+l+ 46.7%

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

      10. +-commutative 46.7%

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

      11. count-2 46.7%

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

      12. fma-neg 46.5%

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

      13. associate-+l+ 46.5%

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

      14. +-commutative 46.5%

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

      15. count-2 46.5%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around 0 57.5%

      \[\leadsto \color{blue}{5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 57.5%

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

   7. Simplified 57.5%

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

   if 2.5 < 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 y around inf 48.9%

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

   6. Taylor expanded in x around inf 48.9%

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

      1. *-commutative 48.9%

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

   8. Simplified 48.9%

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

Alternative 12: 47.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-14}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-39}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.1e-14) (* x t) (if (<= x 2.7e-39) (* y 5.0) (* x (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e-14) {
		tmp = x * t;
	} else if (x <= 2.7e-39) {
		tmp = y * 5.0;
	} else {
		tmp = x * (z * 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 <= (-1.1d-14)) then
        tmp = x * t
    else if (x <= 2.7d-39) then
        tmp = y * 5.0d0
    else
        tmp = x * (z * 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 <= -1.1e-14) {
		tmp = x * t;
	} else if (x <= 2.7e-39) {
		tmp = y * 5.0;
	} else {
		tmp = x * (z * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.1e-14:
		tmp = x * t
	elif x <= 2.7e-39:
		tmp = y * 5.0
	else:
		tmp = x * (z * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.1e-14)
		tmp = Float64(x * t);
	elseif (x <= 2.7e-39)
		tmp = Float64(y * 5.0);
	else
		tmp = Float64(x * Float64(z * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.1e-14)
		tmp = x * t;
	elseif (x <= 2.7e-39)
		tmp = y * 5.0;
	else
		tmp = x * (z * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.1e-14], N[(x * t), $MachinePrecision], If[LessEqual[x, 2.7e-39], N[(y * 5.0), $MachinePrecision], N[(x * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e-14

   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. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. flip-+ 69.8%

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

      4. associate-*r/ 56.0%

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

      5. fma-neg 60.5%

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

      6. associate-+l+ 60.5%

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

      7. +-commutative 60.5%

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

      8. count-2 60.5%

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

      9. associate-+l+ 60.5%

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

      10. +-commutative 60.5%

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

      11. count-2 60.5%

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

      12. fma-neg 56.0%

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

      13. associate-+l+ 56.0%

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

      14. +-commutative 56.0%

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

      15. count-2 56.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around inf 39.6%

      \[\leadsto \color{blue}{t \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 39.6%

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

   7. Simplified 39.6%

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

   if -1.1e-14 < x < 2.7000000000000001e-39

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-define 99.2%

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

      3. flip-+ 46.9%

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

      4. associate-*r/ 46.1%

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

      5. fma-neg 46.3%

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

      6. associate-+l+ 46.3%

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

      7. +-commutative 46.3%

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

      8. count-2 46.3%

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

      9. associate-+l+ 46.3%

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

      10. +-commutative 46.3%

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

      11. count-2 46.3%

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

      12. fma-neg 46.1%

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

      13. associate-+l+ 46.1%

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

      14. +-commutative 46.1%

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

      15. count-2 46.1%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around 0 61.0%

      \[\leadsto \color{blue}{5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 61.0%

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

   7. Simplified 61.0%

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

   if 2.7000000000000001e-39 < 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. flip-+ 80.0%

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

      4. associate-*r/ 66.9%

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

      5. fma-neg 70.2%

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

      6. associate-+l+ 70.2%

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

      7. +-commutative 70.2%

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

      8. count-2 70.2%

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

      9. associate-+l+ 70.2%

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

      10. +-commutative 70.2%

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

      11. count-2 70.2%

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

      12. fma-neg 66.9%

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

      13. associate-+l+ 66.9%

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

      14. +-commutative 66.9%

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

      15. count-2 66.9%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf 37.9%

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

      1. associate-*r* 37.9%

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

      2. *-commutative 37.9%

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

      3. associate-*r* 37.9%

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

   7. Simplified 37.9%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-14}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-39}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 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 99.5%

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

3. Final simplification 99.5%

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

Alternative 14: 47.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-9} \lor \neg \left(x \leq 1.95 \cdot 10^{-38}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.7e-9) (not (<= x 1.95e-38))) (* x t) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.7e-9) || !(x <= 1.95e-38)) {
		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 <= (-2.7d-9)) .or. (.not. (x <= 1.95d-38))) 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 <= -2.7e-9) || !(x <= 1.95e-38)) {
		tmp = x * t;
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.7e-9) or not (x <= 1.95e-38):
		tmp = x * t
	else:
		tmp = y * 5.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.7e-9) || !(x <= 1.95e-38))
		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 <= -2.7e-9) || ~((x <= 1.95e-38)))
		tmp = x * t;
	else
		tmp = y * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.7e-9], N[Not[LessEqual[x, 1.95e-38]], $MachinePrecision]], N[(x * t), $MachinePrecision], N[(y * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002e-9 or 1.95e-38 < 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. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. flip-+ 75.3%

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

      4. associate-*r/ 61.7%

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

      5. fma-neg 65.7%

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

      6. associate-+l+ 65.7%

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

      7. +-commutative 65.7%

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

      8. count-2 65.7%

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

      9. associate-+l+ 65.7%

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

      10. +-commutative 65.7%

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

      11. count-2 65.7%

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

      12. fma-neg 61.7%

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

      13. associate-+l+ 61.7%

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

      14. +-commutative 61.7%

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

      15. count-2 61.7%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around inf 36.8%

      \[\leadsto \color{blue}{t \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 36.8%

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

   7. Simplified 36.8%

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

   if -2.7000000000000002e-9 < x < 1.95e-38

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-define 99.2%

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

      3. flip-+ 46.6%

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

      4. associate-*r/ 45.8%

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

      5. fma-neg 46.0%

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

      6. associate-+l+ 46.0%

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

      7. +-commutative 46.0%

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

      8. count-2 46.0%

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

      9. associate-+l+ 46.0%

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

      10. +-commutative 46.0%

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

      11. count-2 46.0%

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

      12. fma-neg 45.8%

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

      13. associate-+l+ 45.8%

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

      14. +-commutative 45.8%

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

      15. count-2 45.8%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around 0 60.6%

      \[\leadsto \color{blue}{5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 60.6%

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

   7. Simplified 60.6%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-9} \lor \neg \left(x \leq 1.95 \cdot 10^{-38}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ x \cdot t \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. flip-+ 61.0%

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

   4. associate-*r/ 53.8%

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

   5. fma-neg 55.8%

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

   6. associate-+l+ 55.9%

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

   7. +-commutative 55.9%

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

   8. count-2 55.9%

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

   9. associate-+l+ 55.8%

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

   10. +-commutative 55.8%

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

   11. count-2 55.8%

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

   12. fma-neg 53.8%

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

   13. associate-+l+ 53.8%

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

   14. +-commutative 53.8%

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

   15. count-2 53.8%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in t around inf 28.5%

   \[\leadsto \color{blue}{t \cdot x} \]
6. Step-by-step derivation

   1. *-commutative 28.5%

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

7. Simplified 28.5%

   \[\leadsto \color{blue}{x \cdot t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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)))