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

Percentage Accurate: 90.8% → 95.0%

Time: 9.4s

Alternatives: 7

Speedup: 0.7×

• 49.3% 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: 90.8% 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: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+298}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -4\right) \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+298)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* (* z -4.0) (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+298) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (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+298) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (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+298) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (z * -4.0) * (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+298:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (z * -4.0) * (z * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+298)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(z * -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+298)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (z * -4.0) * (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+298], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * -4.0), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000003e298

   1. Initial program 96.4%

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

   if 5.0000000000000003e298 < (*.f64 z z)

   1. Initial program 65.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 92.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+298}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -4\right) \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+203}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -4\right) \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 -1e-107)
     (* y (* 4.0 t))
     (if (<= t_1 5e+203) (* 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 (t_1 <= -1e-107) {
		tmp = y * (4.0 * t);
	} else if (t_1 <= 5e+203) {
		tmp = x * x;
	} else {
		tmp = (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 (t_1 <= (-1d-107)) then
        tmp = y * (4.0d0 * t)
    else if (t_1 <= 5d+203) then
        tmp = x * x
    else
        tmp = (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 (t_1 <= -1e-107) {
		tmp = y * (4.0 * t);
	} else if (t_1 <= 5e+203) {
		tmp = x * x;
	} else {
		tmp = (z * -4.0) * (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= -1e-107:
		tmp = y * (4.0 * t)
	elif t_1 <= 5e+203:
		tmp = x * x
	else:
		tmp = (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 (t_1 <= -1e-107)
		tmp = Float64(y * Float64(4.0 * t));
	elseif (t_1 <= 5e+203)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(z * -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 (t_1 <= -1e-107)
		tmp = y * (4.0 * t);
	elseif (t_1 <= 5e+203)
		tmp = x * x;
	else
		tmp = (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[t$95$1, -1e-107], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+203], N[(x * x), $MachinePrecision], N[(N[(z * -4.0), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -1e-107

   1. Initial program 98.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if -1e-107 < (-.f64 (*.f64 z z) t) < 4.99999999999999994e203

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 4.99999999999999994e203 < (-.f64 (*.f64 z z) t)

   1. Initial program 74.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.6

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

   5. Applied rewrites 63.6%

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

   7. Applied rewrites 75.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -1 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \cdot z - t \leq 5 \cdot 10^{+203}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -4\right) \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+203}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 -1e-107)
     (* y (* 4.0 t))
     (if (<= t_1 5e+203) (* x x) (* -4.0 (* (* z z) y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -1e-107) {
		tmp = y * (4.0 * t);
	} else if (t_1 <= 5e+203) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (z * z) - t
    if (t_1 <= (-1d-107)) then
        tmp = y * (4.0d0 * t)
    else if (t_1 <= 5d+203) 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 t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -1e-107) {
		tmp = y * (4.0 * t);
	} else if (t_1 <= 5e+203) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= -1e-107:
		tmp = y * (4.0 * t)
	elif t_1 <= 5e+203:
		tmp = x * x
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (t_1 <= -1e-107)
		tmp = Float64(y * Float64(4.0 * t));
	elseif (t_1 <= 5e+203)
		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)
	t_1 = (z * z) - t;
	tmp = 0.0;
	if (t_1 <= -1e-107)
		tmp = y * (4.0 * t);
	elseif (t_1 <= 5e+203)
		tmp = x * x;
	else
		tmp = -4.0 * ((z * 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[t$95$1, -1e-107], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+203], N[(x * x), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -1e-107

   1. Initial program 98.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if -1e-107 < (-.f64 (*.f64 z z) t) < 4.99999999999999994e203

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 4.99999999999999994e203 < (-.f64 (*.f64 z z) t)

   1. Initial program 74.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.6

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

   5. Applied rewrites 63.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -1 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \cdot z - t \leq 5 \cdot 10^{+203}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.6 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+217}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot \left(z \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -4\right) \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 6.6e+52)
   (fma y (* 4.0 t) (* x x))
   (if (<= z 1.95e+217)
     (- (* x x) (* z (* y (* z 4.0))))
     (* (* z -4.0) (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 6.6e+52) {
		tmp = fma(y, (4.0 * t), (x * x));
	} else if (z <= 1.95e+217) {
		tmp = (x * x) - (z * (y * (z * 4.0)));
	} else {
		tmp = (z * -4.0) * (z * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 6.6e+52)
		tmp = fma(y, Float64(4.0 * t), Float64(x * x));
	elseif (z <= 1.95e+217)
		tmp = Float64(Float64(x * x) - Float64(z * Float64(y * Float64(z * 4.0))));
	else
		tmp = Float64(Float64(z * -4.0) * Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 6.6e+52], N[(y * N[(4.0 * t), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.95e+217], N[(N[(x * x), $MachinePrecision] - N[(z * N[(y * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * -4.0), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 6.6e52

   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 z around 0

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 4 \cdot t, {x}^{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, {x}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, {x}^{2}\right) \]

      9. unpow2 N/A

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

      10. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if 6.6e52 < z < 1.94999999999999997e217

   1. Initial program 77.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if 1.94999999999999997e217 < z

   1. Initial program 77.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.6 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+217}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot \left(z \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -4\right) \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -4\right) \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1e+97) (fma y (* 4.0 t) (* x x)) (* (* z -4.0) (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1e+97) {
		tmp = fma(y, (4.0 * t), (x * x));
	} else {
		tmp = (z * -4.0) * (z * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1e+97)
		tmp = fma(y, Float64(4.0 * t), Float64(x * x));
	else
		tmp = Float64(Float64(z * -4.0) * Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1e+97], N[(y * N[(4.0 * t), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * -4.0), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.0000000000000001e97

   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 z around 0

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 4 \cdot t, {x}^{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, {x}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, {x}^{2}\right) \]

      9. unpow2 N/A

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

      10. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   if 1.0000000000000001e97 < z

   1. Initial program 75.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 83.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -4\right) \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.6 \cdot 10^{-49}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2.6e-49) (* y (* 4.0 t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2.6e-49) {
		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 * x) <= 2.6d-49) 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 * x) <= 2.6e-49) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 2.6e-49:
		tmp = y * (4.0 * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2.6e-49)
		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 * x) <= 2.6e-49)
		tmp = y * (4.0 * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.6e-49], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.59999999999999995e-49

   1. Initial program 93.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   if 2.59999999999999995e-49 < (*.f64 x x)

   1. Initial program 85.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.6 \cdot 10^{-49}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.7% accurate, 4.5× 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 89.0%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 41.8

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

5. Applied rewrites 41.8%

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

Developer Target 1: 90.9% 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 2024221 
(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))))