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

Percentage Accurate: 90.9% → 94.7%

Time: 7.7s

Alternatives: 6

Speedup: 0.9×

• 50.0% 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: 90.9% 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: 94.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{+161}:\\ \;\;\;\;x \cdot x - y \cdot \left(4 \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 8.2e+161)
   (- (* x x) (* y (* 4.0 (- (* z z) t))))
   (* z (* -4.0 (* z y)))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8.2e+161) {
		tmp = (x * x) - (y * (4.0 * ((z * z) - t)));
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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 <= 8.2d+161) then
        tmp = (x * x) - (y * (4.0d0 * ((z * z) - t)))
    else
        tmp = z * ((-4.0d0) * (z * y))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8.2e+161) {
		tmp = (x * x) - (y * (4.0 * ((z * z) - t)));
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if z <= 8.2e+161:
		tmp = (x * x) - (y * (4.0 * ((z * z) - t)))
	else:
		tmp = z * (-4.0 * (z * y))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 8.2e+161)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(4.0 * Float64(Float64(z * z) - t))));
	else
		tmp = Float64(z * Float64(-4.0 * Float64(z * y)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 8.2e+161)
		tmp = (x * x) - (y * (4.0 * ((z * z) - t)));
	else
		tmp = z * (-4.0 * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 8.2e+161], N[(N[(x * x), $MachinePrecision] - N[(y * N[(4.0 * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.2000000000000002e161

   1. Initial program 93.5%

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

      1. associate-*l* 93.9%

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

   3. Simplified 93.9%

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

   if 8.2000000000000002e161 < z

   1. Initial program 59.7%

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

      1. associate-*l* 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in z around inf 63.6%

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

      1. unpow2 63.6%

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

      2. associate-*r* 63.6%

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

      3. *-commutative 63.6%

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

      4. associate-*r* 92.3%

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

      5. *-commutative 92.3%

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

      6. *-commutative 92.3%

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

      7. associate-*l* 92.3%

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

   6. Simplified 92.3%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{+161}:\\ \;\;\;\;x \cdot x - y \cdot \left(4 \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 2: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+45}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+162}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.45e+45)
   (- (* x x) (* t (* y -4.0)))
   (if (<= z 1.02e+162) (* (- (* z z) t) (* y -4.0)) (* z (* -4.0 (* z y))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.45e+45) {
		tmp = (x * x) - (t * (y * -4.0));
	} else if (z <= 1.02e+162) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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 <= 1.45d+45) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else if (z <= 1.02d+162) then
        tmp = ((z * z) - t) * (y * (-4.0d0))
    else
        tmp = z * ((-4.0d0) * (z * y))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.45e+45) {
		tmp = (x * x) - (t * (y * -4.0));
	} else if (z <= 1.02e+162) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if z <= 1.45e+45:
		tmp = (x * x) - (t * (y * -4.0))
	elif z <= 1.02e+162:
		tmp = ((z * z) - t) * (y * -4.0)
	else:
		tmp = z * (-4.0 * (z * y))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.45e+45)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	elseif (z <= 1.02e+162)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0));
	else
		tmp = Float64(z * Float64(-4.0 * Float64(z * y)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.45e+45)
		tmp = (x * x) - (t * (y * -4.0));
	elseif (z <= 1.02e+162)
		tmp = ((z * z) - t) * (y * -4.0);
	else
		tmp = z * (-4.0 * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 1.45e+45], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.02e+162], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], N[(z * N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.4499999999999999e45

   1. Initial program 93.5%

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

      1. associate-*l* 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in z around 0 78.5%

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

      1. associate-*r* 78.5%

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

      2. *-commutative 78.5%

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

   6. Simplified 78.5%

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

   if 1.4499999999999999e45 < z < 1.01999999999999993e162

   1. Initial program 94.2%

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

      1. associate-*l* 94.2%

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

   3. Simplified 94.2%

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

      1. sub-neg 94.2%

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

      2. +-commutative 94.2%

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

      3. distribute-rgt-neg-in 94.2%

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

      4. distribute-lft-neg-in 94.2%

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

      5. metadata-eval 94.2%

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

      6. associate-*l* 94.2%

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

      7. *-commutative 94.2%

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

      8. add-cube-cbrt 93.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{z \cdot z - t} \cdot \sqrt[3]{z \cdot z - t}\right) \cdot \sqrt[3]{z \cdot z - t}\right)} \cdot \left(y \cdot -4\right) + x \cdot x \]

      9. associate-*l* 93.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{z \cdot z - t} \cdot \sqrt[3]{z \cdot z - t}\right) \cdot \left(\sqrt[3]{z \cdot z - t} \cdot \left(y \cdot -4\right)\right)} + x \cdot x \]

      10. fma-def 93.8%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{z \cdot z - t} \cdot \sqrt[3]{z \cdot z - t}, \sqrt[3]{z \cdot z - t} \cdot \left(y \cdot -4\right), x \cdot x\right)} \]

      11. pow2 93.8%

          \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{z \cdot z - t}\right)}^{2}}, \sqrt[3]{z \cdot z - t} \cdot \left(y \cdot -4\right), x \cdot x\right) \]

   5. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{z \cdot z - t}\right)}^{2}, \sqrt[3]{z \cdot z - t} \cdot \left(y \cdot -4\right), x \cdot x\right)} \]

   6. Taylor expanded in y around inf 67.3%

      \[\leadsto \color{blue}{-4 \cdot \left(\left(\left({z}^{2} - t\right) \cdot y\right) \cdot {1}^{0.3333333333333333}\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 67.3%

         \[\leadsto -4 \cdot \left(\left(\left({z}^{2} - t\right) \cdot y\right) \cdot \color{blue}{1}\right) \]

      2. associate-*r* 67.3%

         \[\leadsto \color{blue}{\left(-4 \cdot \left(\left({z}^{2} - t\right) \cdot y\right)\right) \cdot 1} \]

      3. *-rgt-identity 67.3%

         \[\leadsto \left(-4 \cdot \left(\color{blue}{\left(\left({z}^{2} - t\right) \cdot 1\right)} \cdot y\right)\right) \cdot 1 \]

      4. *-rgt-identity 67.3%

         \[\leadsto \left(-4 \cdot \left(\color{blue}{\left({z}^{2} - t\right)} \cdot y\right)\right) \cdot 1 \]

      5. unpow2 67.3%

         \[\leadsto \left(-4 \cdot \left(\left(\color{blue}{z \cdot z} - t\right) \cdot y\right)\right) \cdot 1 \]

      6. *-commutative 67.3%

         \[\leadsto \color{blue}{\left(\left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\right)} \cdot 1 \]

      7. associate-*r* 67.3%

         \[\leadsto \color{blue}{\left(\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)} \cdot 1 \]

      8. *-rgt-identity 67.3%

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

   8. Simplified 67.3%

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

   if 1.01999999999999993e162 < z

   1. Initial program 59.7%

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

      1. associate-*l* 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in z around inf 63.6%

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

      1. unpow2 63.6%

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

      2. associate-*r* 63.6%

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

      3. *-commutative 63.6%

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

      4. associate-*r* 92.3%

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

      5. *-commutative 92.3%

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

      6. *-commutative 92.3%

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

      7. associate-*l* 92.3%

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

   6. Simplified 92.3%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+45}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+162}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.8 \cdot 10^{+194}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 4.8e+194) (* (- (* z z) t) (* y -4.0)) (* x x)))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 4.8e+194) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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) <= 4.8d+194) then
        tmp = ((z * z) - t) * (y * (-4.0d0))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 4.8e+194) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 4.8e+194:
		tmp = ((z * z) - t) * (y * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 4.8e+194)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 4.8e+194)
		tmp = ((z * z) - t) * (y * -4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 4.8e+194], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.8e194

   1. Initial program 94.4%

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

      1. associate-*l* 94.9%

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

   3. Simplified 94.9%

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

      1. sub-neg 94.9%

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

      2. +-commutative 94.9%

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

      3. distribute-rgt-neg-in 94.9%

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

      4. distribute-lft-neg-in 94.9%

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

      5. metadata-eval 94.9%

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

      6. associate-*l* 94.4%

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

      7. *-commutative 94.4%

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

      8. add-cube-cbrt 93.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{z \cdot z - t} \cdot \sqrt[3]{z \cdot z - t}\right) \cdot \sqrt[3]{z \cdot z - t}\right)} \cdot \left(y \cdot -4\right) + x \cdot x \]

      9. associate-*l* 93.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{z \cdot z - t} \cdot \sqrt[3]{z \cdot z - t}\right) \cdot \left(\sqrt[3]{z \cdot z - t} \cdot \left(y \cdot -4\right)\right)} + x \cdot x \]

      10. fma-def 93.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{z \cdot z - t} \cdot \sqrt[3]{z \cdot z - t}, \sqrt[3]{z \cdot z - t} \cdot \left(y \cdot -4\right), x \cdot x\right)} \]

      11. pow2 93.6%

          \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{z \cdot z - t}\right)}^{2}}, \sqrt[3]{z \cdot z - t} \cdot \left(y \cdot -4\right), x \cdot x\right) \]

   5. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{z \cdot z - t}\right)}^{2}, \sqrt[3]{z \cdot z - t} \cdot \left(y \cdot -4\right), x \cdot x\right)} \]

   6. Taylor expanded in y around inf 82.1%

      \[\leadsto \color{blue}{-4 \cdot \left(\left(\left({z}^{2} - t\right) \cdot y\right) \cdot {1}^{0.3333333333333333}\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 82.1%

         \[\leadsto -4 \cdot \left(\left(\left({z}^{2} - t\right) \cdot y\right) \cdot \color{blue}{1}\right) \]

      2. associate-*r* 82.1%

         \[\leadsto \color{blue}{\left(-4 \cdot \left(\left({z}^{2} - t\right) \cdot y\right)\right) \cdot 1} \]

      3. *-rgt-identity 82.1%

         \[\leadsto \left(-4 \cdot \left(\color{blue}{\left(\left({z}^{2} - t\right) \cdot 1\right)} \cdot y\right)\right) \cdot 1 \]

      4. *-rgt-identity 82.1%

         \[\leadsto \left(-4 \cdot \left(\color{blue}{\left({z}^{2} - t\right)} \cdot y\right)\right) \cdot 1 \]

      5. unpow2 82.1%

         \[\leadsto \left(-4 \cdot \left(\left(\color{blue}{z \cdot z} - t\right) \cdot y\right)\right) \cdot 1 \]

      6. *-commutative 82.1%

         \[\leadsto \color{blue}{\left(\left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\right)} \cdot 1 \]

      7. associate-*r* 81.6%

         \[\leadsto \color{blue}{\left(\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)} \cdot 1 \]

      8. *-rgt-identity 81.6%

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

   8. Simplified 81.6%

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

   if 4.8e194 < (*.f64 x x)

   1. Initial program 81.9%

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

      1. associate-*l* 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in x around inf 90.0%

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

      1. unpow2 90.0%

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

   6. Simplified 90.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.8 \cdot 10^{+194}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 61.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{-118}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+49}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 4.5e-118)
   (* t (* y 4.0))
   (if (<= z 2.6e+49) (* x x) (* z (* -4.0 (* z y))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.5e-118) {
		tmp = t * (y * 4.0);
	} else if (z <= 2.6e+49) {
		tmp = x * x;
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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 <= 4.5d-118) then
        tmp = t * (y * 4.0d0)
    else if (z <= 2.6d+49) then
        tmp = x * x
    else
        tmp = z * ((-4.0d0) * (z * y))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.5e-118) {
		tmp = t * (y * 4.0);
	} else if (z <= 2.6e+49) {
		tmp = x * x;
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if z <= 4.5e-118:
		tmp = t * (y * 4.0)
	elif z <= 2.6e+49:
		tmp = x * x
	else:
		tmp = z * (-4.0 * (z * y))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 4.5e-118)
		tmp = Float64(t * Float64(y * 4.0));
	elseif (z <= 2.6e+49)
		tmp = Float64(x * x);
	else
		tmp = Float64(z * Float64(-4.0 * Float64(z * y)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 4.5e-118)
		tmp = t * (y * 4.0);
	elseif (z <= 2.6e+49)
		tmp = x * x;
	else
		tmp = z * (-4.0 * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 4.5e-118], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e+49], N[(x * x), $MachinePrecision], N[(z * N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 4.5e-118

   1. Initial program 92.1%

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

      1. associate-*l* 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in t around inf 41.5%

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

      1. associate-*r* 41.5%

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

   6. Simplified 41.5%

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

   if 4.5e-118 < z < 2.59999999999999989e49

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 61.8%

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

      1. unpow2 61.8%

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

   6. Simplified 61.8%

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

   if 2.59999999999999989e49 < z

   1. Initial program 73.8%

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

      1. associate-*l* 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in z around inf 63.1%

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

      1. unpow2 63.1%

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

      2. associate-*r* 63.1%

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

      3. *-commutative 63.1%

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

      4. associate-*r* 77.9%

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

      5. *-commutative 77.9%

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

      6. *-commutative 77.9%

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

      7. associate-*l* 77.9%

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

   6. Simplified 77.9%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{-118}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+49}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 5: 58.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.55 \cdot 10^{+110}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.55e+110) (* t (* y 4.0)) (* x x)))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.55e+110) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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) <= 1.55d+110) then
        tmp = t * (y * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.55e+110) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.55e+110:
		tmp = t * (y * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.55e+110)
		tmp = Float64(t * Float64(y * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1.55e+110)
		tmp = t * (y * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.55e+110], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.55000000000000009e110

   1. Initial program 93.6%

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

      1. associate-*l* 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in t around inf 52.0%

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

      1. associate-*r* 52.0%

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

   6. Simplified 52.0%

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

   if 1.55000000000000009e110 < (*.f64 x x)

   1. Initial program 85.3%

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

      1. associate-*l* 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in x around inf 80.4%

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

      1. unpow2 80.4%

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

   6. Simplified 80.4%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.55 \cdot 10^{+110}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 41.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ x \cdot x \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t) :precision binary64 (* x x))

C

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

Fortran

NOTE: z should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := N[(x * x), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

   1. associate-*l* 90.4%

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

3. Simplified 90.4%

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

4. Taylor expanded in x around inf 41.5%

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

   1. unpow2 41.5%

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

6. Simplified 41.5%

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

7. Final simplification 41.5%

   \[\leadsto x \cdot x \]

Developer target: 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 2023230 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

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