Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B

Percentage Accurate: 91.0% → 95.6%

Time: 11.7s

Alternatives: 10

Speedup: 0.6×

• 48.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - 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 * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - 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 * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z + t\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+228}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t\_1 \cdot \frac{t - z \cdot z}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* z z) t)))
   (if (<= (* z z) 5e+228)
     (+ (* x x) (* (* y 4.0) (* t_1 (/ (- t (* z z)) t_1))))
     (- (* x x) (* z (* 4.0 (* z y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) + t;
	double tmp;
	if ((z * z) <= 5e+228) {
		tmp = (x * x) + ((y * 4.0) * (t_1 * ((t - (z * z)) / t_1)));
	} else {
		tmp = (x * x) - (z * (4.0 * (z * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) + t
    if ((z * z) <= 5d+228) then
        tmp = (x * x) + ((y * 4.0d0) * (t_1 * ((t - (z * z)) / t_1)))
    else
        tmp = (x * x) - (z * (4.0d0 * (z * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) + t;
	double tmp;
	if ((z * z) <= 5e+228) {
		tmp = (x * x) + ((y * 4.0) * (t_1 * ((t - (z * z)) / t_1)));
	} else {
		tmp = (x * x) - (z * (4.0 * (z * y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) + t
	tmp = 0
	if (z * z) <= 5e+228:
		tmp = (x * x) + ((y * 4.0) * (t_1 * ((t - (z * z)) / t_1)))
	else:
		tmp = (x * x) - (z * (4.0 * (z * y)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) + t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+228)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t_1 * Float64(Float64(t - Float64(z * z)) / t_1))));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(4.0 * Float64(z * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) + t;
	tmp = 0.0;
	if ((z * z) <= 5e+228)
		tmp = (x * x) + ((y * 4.0) * (t_1 * ((t - (z * z)) / t_1)));
	else
		tmp = (x * x) - (z * (4.0 * (z * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 5e+228], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t$95$1 * N[(N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e228

   1. Initial program 98.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}{\color{blue}{z \cdot z + t}}\right)\right)\right) \]

      2. difference-of-squares N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\left(z \cdot z + t\right) \cdot \left(z \cdot z - t\right)}{\color{blue}{z \cdot z} + t}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\left(z \cdot z + t\right) \cdot \color{blue}{\frac{z \cdot z - t}{z \cdot z + t}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(\left(z \cdot z + t\right), \color{blue}{\left(\frac{z \cdot z - t}{z \cdot z + t}\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(z \cdot z\right), t\right), \left(\frac{\color{blue}{z \cdot z - t}}{z \cdot z + t}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right), \left(\frac{\color{blue}{z \cdot z} - t}{z \cdot z + t}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right), \mathsf{/.f64}\left(\left(z \cdot z - t\right), \color{blue}{\left(z \cdot z + t\right)}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), t\right), \left(\color{blue}{z \cdot z} + t\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right), \left(\color{blue}{z} \cdot z + t\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right), \mathsf{+.f64}\left(\left(z \cdot z\right), \color{blue}{t}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 98.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 98.8%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(\left(z \cdot z + t\right) \cdot \frac{z \cdot z - t}{z \cdot z + t}\right)} \]

   if 5e228 < (*.f64 z z)

   1. Initial program 74.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \color{blue}{\left({z}^{2}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      2. *-lowering-*.f64 74.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 74.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\left(y \cdot 4\right) \cdot z\right) \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(\left(y \cdot 4\right) \cdot z\right), \color{blue}{z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(\left(4 \cdot y\right) \cdot z\right), z\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(4 \cdot \left(y \cdot z\right)\right), z\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(y \cdot z\right)\right), z\right)\right) \]

      6. *-lowering-*.f64 94.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, z\right)\right), z\right)\right) \]

   7. Applied egg-rr 94.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+228}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(\left(z \cdot z + t\right) \cdot \frac{t - z \cdot z}{z \cdot z + t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 51.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-180}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+90}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 5.4e-180)
   (* x x)
   (if (<= z 5.2e-138)
     (* y (* 4.0 t))
     (if (<= z 1.45e+90) (* x x) (* z (* z (* y -4.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.4e-180) {
		tmp = x * x;
	} else if (z <= 5.2e-138) {
		tmp = y * (4.0 * t);
	} else if (z <= 1.45e+90) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	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 (z <= 5.4d-180) then
        tmp = x * x
    else if (z <= 5.2d-138) then
        tmp = y * (4.0d0 * t)
    else if (z <= 1.45d+90) then
        tmp = x * x
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.4e-180) {
		tmp = x * x;
	} else if (z <= 5.2e-138) {
		tmp = y * (4.0 * t);
	} else if (z <= 1.45e+90) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 5.4e-180:
		tmp = x * x
	elif z <= 5.2e-138:
		tmp = y * (4.0 * t)
	elif z <= 1.45e+90:
		tmp = x * x
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 5.4e-180)
		tmp = Float64(x * x);
	elseif (z <= 5.2e-138)
		tmp = Float64(y * Float64(4.0 * t));
	elseif (z <= 1.45e+90)
		tmp = Float64(x * x);
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 5.4e-180)
		tmp = x * x;
	elseif (z <= 5.2e-138)
		tmp = y * (4.0 * t);
	elseif (z <= 1.45e+90)
		tmp = x * x;
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 5.4e-180], N[(x * x), $MachinePrecision], If[LessEqual[z, 5.2e-138], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.45e+90], N[(x * x), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.40000000000000028e-180 or 5.2e-138 < z < 1.4500000000000001e90

   1. Initial program 95.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 44.7%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 44.7%

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

   if 5.40000000000000028e-180 < z < 5.2e-138

   1. Initial program 100.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

         \[\leadsto \left(4 \cdot t\right) \cdot \color{blue}{y} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(4 \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(4 \cdot t\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(t \cdot \color{blue}{4}\right)\right) \]

      5. *-lowering-*.f64 63.7%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(t, \color{blue}{4}\right)\right) \]

   5. Simplified 63.7%

      \[\leadsto \color{blue}{y \cdot \left(t \cdot 4\right)} \]

   if 1.4500000000000001e90 < z

   1. Initial program 69.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 64.9%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 64.9%

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

      1. associate-*r* N/A

         \[\leadsto \left(-4 \cdot y\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right) \cdot \left(z \cdot z\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(4 \cdot y\right)\right) \cdot \left(\color{blue}{z} \cdot z\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right) \cdot \color{blue}{z} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right), \color{blue}{z}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right), z\right), z\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(4\right)\right)\right), z\right), z\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot -4\right), z\right), z\right) \]

      10. *-lowering-*.f64 81.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, -4\right), z\right), z\right) \]

   7. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\left(\left(y \cdot -4\right) \cdot z\right) \cdot z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-180}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+90}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.02 \cdot 10^{-178}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+90}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.02e-178)
   (* x x)
   (if (<= z 5.4e-138)
     (* y (* 4.0 t))
     (if (<= z 1.45e+90) (* x x) (* -4.0 (* (* z z) y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.02e-178) {
		tmp = x * x;
	} else if (z <= 5.4e-138) {
		tmp = y * (4.0 * t);
	} else if (z <= 1.45e+90) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.02d-178) then
        tmp = x * x
    else if (z <= 5.4d-138) then
        tmp = y * (4.0d0 * t)
    else if (z <= 1.45d+90) then
        tmp = x * x
    else
        tmp = (-4.0d0) * ((z * z) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.02e-178) {
		tmp = x * x;
	} else if (z <= 5.4e-138) {
		tmp = y * (4.0 * t);
	} else if (z <= 1.45e+90) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.02e-178:
		tmp = x * x
	elif z <= 5.4e-138:
		tmp = y * (4.0 * t)
	elif z <= 1.45e+90:
		tmp = x * x
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.02e-178)
		tmp = Float64(x * x);
	elseif (z <= 5.4e-138)
		tmp = Float64(y * Float64(4.0 * t));
	elseif (z <= 1.45e+90)
		tmp = Float64(x * x);
	else
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.02e-178)
		tmp = x * x;
	elseif (z <= 5.4e-138)
		tmp = y * (4.0 * t);
	elseif (z <= 1.45e+90)
		tmp = x * x;
	else
		tmp = -4.0 * ((z * z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.02e-178], N[(x * x), $MachinePrecision], If[LessEqual[z, 5.4e-138], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.45e+90], N[(x * x), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.02000000000000006e-178 or 5.40000000000000057e-138 < z < 1.4500000000000001e90

   1. Initial program 95.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 44.7%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 44.7%

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

   if 1.02000000000000006e-178 < z < 5.40000000000000057e-138

   1. Initial program 100.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

         \[\leadsto \left(4 \cdot t\right) \cdot \color{blue}{y} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(4 \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(4 \cdot t\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(t \cdot \color{blue}{4}\right)\right) \]

      5. *-lowering-*.f64 63.7%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(t, \color{blue}{4}\right)\right) \]

   5. Simplified 63.7%

      \[\leadsto \color{blue}{y \cdot \left(t \cdot 4\right)} \]

   if 1.4500000000000001e90 < z

   1. Initial program 69.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 64.9%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 64.9%

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot z\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.02 \cdot 10^{-178}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+90}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+228}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+228)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (- (* x x) (* z (* 4.0 (* z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+228) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (4.0 * (z * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 5d+228) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - (z * (4.0d0 * (z * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+228) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (4.0 * (z * y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+228:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - (z * (4.0 * (z * y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+228)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(4.0 * Float64(z * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e+228)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - (z * (4.0 * (z * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+228], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e228

   1. Initial program 98.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   if 5e228 < (*.f64 z z)

   1. Initial program 74.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \color{blue}{\left({z}^{2}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      2. *-lowering-*.f64 74.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 74.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\left(y \cdot 4\right) \cdot z\right) \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(\left(y \cdot 4\right) \cdot z\right), \color{blue}{z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(\left(4 \cdot y\right) \cdot z\right), z\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(4 \cdot \left(y \cdot z\right)\right), z\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(y \cdot z\right)\right), z\right)\right) \]

      6. *-lowering-*.f64 94.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, z\right)\right), z\right)\right) \]

   7. Applied egg-rr 94.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+228}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-13)
   (- (* x x) (* -4.0 (* y t)))
   (- (* x x) (* z (* 4.0 (* z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-13) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = (x * x) - (z * (4.0 * (z * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 5d-13) then
        tmp = (x * x) - ((-4.0d0) * (y * t))
    else
        tmp = (x * x) - (z * (4.0d0 * (z * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-13) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = (x * x) - (z * (4.0 * (z * y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e-13:
		tmp = (x * x) - (-4.0 * (y * t))
	else:
		tmp = (x * x) - (z * (4.0 * (z * y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e-13)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(y * t)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(4.0 * Float64(z * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e-13)
		tmp = (x * x) - (-4.0 * (y * t));
	else
		tmp = (x * x) - (z * (4.0 * (z * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-13], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999999e-13

   1. Initial program 100.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(-4 \cdot \left(t \cdot y\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \color{blue}{\left(t \cdot y\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 94.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   5. Simplified 94.0%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(y \cdot t\right)} \]

   if 4.9999999999999999e-13 < (*.f64 z z)

   1. Initial program 83.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \color{blue}{\left({z}^{2}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      2. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 78.3%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\left(y \cdot 4\right) \cdot z\right) \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(\left(y \cdot 4\right) \cdot z\right), \color{blue}{z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(\left(4 \cdot y\right) \cdot z\right), z\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(4 \cdot \left(y \cdot z\right)\right), z\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(y \cdot z\right)\right), z\right)\right) \]

      6. *-lowering-*.f64 89.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, z\right)\right), z\right)\right) \]

   7. Applied egg-rr 89.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+40}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(y \cdot \left(z - \frac{t}{z}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+40)
   (- (* x x) (* -4.0 (* y t)))
   (* -4.0 (* z (* y (- z (/ t z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+40) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = -4.0 * (z * (y * (z - (t / 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 * z) <= 5d+40) then
        tmp = (x * x) - ((-4.0d0) * (y * t))
    else
        tmp = (-4.0d0) * (z * (y * (z - (t / z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+40) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = -4.0 * (z * (y * (z - (t / z))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+40:
		tmp = (x * x) - (-4.0 * (y * t))
	else:
		tmp = -4.0 * (z * (y * (z - (t / z))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+40)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(y * t)));
	else
		tmp = Float64(-4.0 * Float64(z * Float64(y * Float64(z - Float64(t / z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e+40)
		tmp = (x * x) - (-4.0 * (y * t));
	else
		tmp = -4.0 * (z * (y * (z - (t / z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+40], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(z * N[(y * N[(z - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000003e40

   1. Initial program 99.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(-4 \cdot \left(t \cdot y\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \color{blue}{\left(t \cdot y\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   5. Simplified 92.5%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(y \cdot t\right)} \]

   if 5.00000000000000003e40 < (*.f64 z z)

   1. Initial program 82.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 4\right) \cdot \frac{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}{\color{blue}{z \cdot z + t}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 4\right) \cdot \frac{1}{\color{blue}{\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}}}\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{y \cdot 4}{\color{blue}{\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\left(y \cdot 4\right), \color{blue}{\left(\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\color{blue}{z \cdot z + t}}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{1}{\color{blue}{\frac{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}{z \cdot z + t}}}\right)\right)\right) \]

      7. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{1}{z \cdot z - \color{blue}{t}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \color{blue}{\left(z \cdot z - t\right)}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(z \cdot z\right), \color{blue}{t}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 82.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right)\right)\right)\right) \]

   4. Applied egg-rr 82.2%

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{z \cdot z - t}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot \left({z}^{2} - t\right)\right) \cdot \color{blue}{-4} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left({z}^{2} - t\right)\right), \color{blue}{-4}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left({z}^{2} - t\right)\right), -4\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left({z}^{2}\right), t\right)\right), -4\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(z \cdot z\right), t\right)\right), -4\right) \]

      6. *-lowering-*.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right)\right), -4\right) \]

   7. Simplified 72.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(z \cdot z - t\right)\right) \cdot -4} \]

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({z}^{2} \cdot \left(y + -1 \cdot \frac{t \cdot y}{{z}^{2}}\right)\right)}, -4\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(z \cdot z\right) \cdot \left(y + -1 \cdot \frac{t \cdot y}{{z}^{2}}\right)\right), -4\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \left(z \cdot \left(y + -1 \cdot \frac{t \cdot y}{{z}^{2}}\right)\right)\right), -4\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot \left(y + -1 \cdot \frac{t \cdot y}{{z}^{2}}\right)\right)\right), -4\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(y + -1 \cdot \frac{t \cdot y}{{z}^{2}}\right)\right)\right), -4\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(y + \left(\mathsf{neg}\left(\frac{t \cdot y}{{z}^{2}}\right)\right)\right)\right)\right), -4\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(y - \frac{t \cdot y}{{z}^{2}}\right)\right)\right), -4\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \left(\frac{t \cdot y}{{z}^{2}}\right)\right)\right)\right), -4\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \left(\frac{y \cdot t}{{z}^{2}}\right)\right)\right)\right), -4\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \left(y \cdot \frac{t}{{z}^{2}}\right)\right)\right)\right), -4\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{t}{{z}^{2}}\right)\right)\right)\right)\right), -4\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(t, \left({z}^{2}\right)\right)\right)\right)\right)\right), -4\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right)\right)\right)\right), -4\right) \]

      13. *-lowering-*.f64 82.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right)\right), -4\right) \]

   10. Simplified 82.3%

       \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot \left(y - y \cdot \frac{t}{z \cdot z}\right)\right)\right)} \cdot -4 \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{\left(\frac{-1 \cdot \left(t \cdot y\right) + y \cdot {z}^{2}}{z}\right)}\right), -4\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\left(-1 \cdot t\right) \cdot y + y \cdot {z}^{2}}{z}\right)\right), -4\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\left(-1 \cdot t\right) \cdot y + {z}^{2} \cdot y}{z}\right)\right), -4\right) \]

       3. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{y \cdot \left(-1 \cdot t + {z}^{2}\right)}{z}\right)\right), -4\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(y \cdot \frac{-1 \cdot t + {z}^{2}}{z}\right)\right), -4\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(y \cdot \frac{{z}^{2} + -1 \cdot t}{z}\right)\right), -4\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(\frac{{z}^{2} + -1 \cdot t}{z}\right)\right)\right), -4\right) \]

       7. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(\frac{{z}^{2} + \left(\mathsf{neg}\left(t\right)\right)}{z}\right)\right)\right), -4\right) \]

       8. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(\frac{{z}^{2} - t}{z}\right)\right)\right), -4\right) \]

       9. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(\frac{{z}^{2}}{z} - \frac{t}{z}\right)\right)\right), -4\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(\frac{z \cdot z}{z} - \frac{t}{z}\right)\right)\right), -4\right) \]

       11. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(z \cdot \frac{z}{z} - \frac{t}{z}\right)\right)\right), -4\right) \]

       12. *-inverses N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(z \cdot 1 - \frac{t}{z}\right)\right)\right), -4\right) \]

       13. *-rgt-identity N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(z - \frac{t}{z}\right)\right)\right), -4\right) \]

       14. --lowering--.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(z, \left(\frac{t}{z}\right)\right)\right)\right), -4\right) \]

       15. /-lowering-/.f64 82.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(z, \mathsf{/.f64}\left(t, z\right)\right)\right)\right), -4\right) \]

   13. Simplified 82.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+40}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(y \cdot \left(z - \frac{t}{z}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+180}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+180) (- (* x x) (* -4.0 (* y t))) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+180) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	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 ((z * z) <= 2d+180) then
        tmp = (x * x) - ((-4.0d0) * (y * t))
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+180) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+180:
		tmp = (x * x) - (-4.0 * (y * t))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+180)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(y * t)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e+180)
		tmp = (x * x) - (-4.0 * (y * t));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+180], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e180

   1. Initial program 98.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(-4 \cdot \left(t \cdot y\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \color{blue}{\left(t \cdot y\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 85.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   5. Simplified 85.8%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(y \cdot t\right)} \]

   if 2e180 < (*.f64 z z)

   1. Initial program 77.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 75.5%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 75.5%

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

      1. associate-*r* N/A

         \[\leadsto \left(-4 \cdot y\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right) \cdot \left(z \cdot z\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(4 \cdot y\right)\right) \cdot \left(\color{blue}{z} \cdot z\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right) \cdot \color{blue}{z} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right), \color{blue}{z}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right), z\right), z\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(4\right)\right)\right), z\right), z\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot -4\right), z\right), z\right) \]

      10. *-lowering-*.f64 89.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, -4\right), z\right), z\right) \]

   7. Applied egg-rr 89.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+180}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+201}:\\ \;\;\;\;-4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2e+201) (* -4.0 (* y (- (* z z) t))) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+201) {
		tmp = -4.0 * (y * ((z * z) - t));
	} else {
		tmp = x * 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 ((x * x) <= 2d+201) then
        tmp = (-4.0d0) * (y * ((z * z) - t))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+201) {
		tmp = -4.0 * (y * ((z * z) - t));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 2e+201:
		tmp = -4.0 * (y * ((z * z) - t))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2e+201)
		tmp = Float64(-4.0 * Float64(y * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 2e+201)
		tmp = -4.0 * (y * ((z * z) - t));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+201], N[(-4.0 * N[(y * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000008e201

   1. Initial program 91.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2} - t\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left({z}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(z \cdot z\right), t\right)\right)\right) \]

      5. *-lowering-*.f64 74.8%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right)\right)\right) \]

   5. Simplified 74.8%

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)} \]

   if 2.00000000000000008e201 < (*.f64 x x)

   1. Initial program 91.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 84.7%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 84.7%

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

Alternative 9: 44.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 4.5e-77) (* y (* 4.0 t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 4.5e-77) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * 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 (x <= 4.5d-77) then
        tmp = y * (4.0d0 * t)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 4.5e-77) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 4.5e-77:
		tmp = y * (4.0 * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 4.5e-77)
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 4.5e-77)
		tmp = y * (4.0 * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 4.5e-77], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5000000000000001e-77

   1. Initial program 91.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

         \[\leadsto \left(4 \cdot t\right) \cdot \color{blue}{y} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(4 \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(4 \cdot t\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(t \cdot \color{blue}{4}\right)\right) \]

      5. *-lowering-*.f64 34.3%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(t, \color{blue}{4}\right)\right) \]

   5. Simplified 34.3%

      \[\leadsto \color{blue}{y \cdot \left(t \cdot 4\right)} \]

   if 4.5000000000000001e-77 < x

   1. Initial program 92.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 65.5%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 65.5%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{{x}^{2}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. *-lowering-*.f64 40.6%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

5. Simplified 40.6%

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

Developer Target 1: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* 4.0 (* y (- (* z z) t)))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - 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 * x) - (4.0d0 * (y * ((z * z) - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - t)));
}

Python

def code(x, y, z, t):
	return (x * x) - (4.0 * (y * ((z * z) - t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(4.0 * Float64(y * Float64(Float64(z * z) - t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - (4.0 * (y * ((z * z) - t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(4.0 * N[(y * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024161 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))