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

Percentage Accurate: 89.9% → 93.6%

Time: 10.2s

Alternatives: 7

Speedup: 0.9×

• 50.2% 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: 89.9% 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: 93.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 7.6 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 7.6e+195)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (- (* x x) (* z (* z (* y 4.0))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7.6e+195) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 7.6e+195)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 7.6e+195], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.6e195

   1. Initial program 94.0%

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

      1. fma-neg 96.2%

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

      2. distribute-lft-neg-in 96.2%

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

      3. *-commutative 96.2%

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

      4. distribute-rgt-neg-in 96.2%

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

      5. metadata-eval 96.2%

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

   3. Simplified 96.2%

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

   if 7.6e195 < z

   1. Initial program 77.7%

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

      1. add-cube-cbrt 77.7%

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

      2. pow3 77.7%

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

      3. pow2 77.7%

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

   3. Applied egg-rr 77.7%

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

   4. Taylor expanded in t around 0 77.7%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot {\color{blue}{\left({\left({z}^{2}\right)}^{0.3333333333333333}\right)}}^{3} \]
   5. Step-by-step derivation

      1. unpow1/3 77.7%

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

   6. Simplified 77.7%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot {\color{blue}{\left(\sqrt[3]{{z}^{2}}\right)}}^{3} \]
   7. Step-by-step derivation

      1. remove-double-div 77.7%

         \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{1}{y \cdot 4}}} \cdot {\left(\sqrt[3]{{z}^{2}}\right)}^{3} \]

      2. rem-cube-cbrt 77.7%

         \[\leadsto x \cdot x - \frac{1}{\frac{1}{y \cdot 4}} \cdot \color{blue}{{z}^{2}} \]

      3. associate-*l/ 77.7%

         \[\leadsto x \cdot x - \color{blue}{\frac{1 \cdot {z}^{2}}{\frac{1}{y \cdot 4}}} \]

      4. *-un-lft-identity 77.7%

         \[\leadsto x \cdot x - \frac{\color{blue}{{z}^{2}}}{\frac{1}{y \cdot 4}} \]

      5. *-commutative 77.7%

         \[\leadsto x \cdot x - \frac{{z}^{2}}{\frac{1}{\color{blue}{4 \cdot y}}} \]

      6. associate-/r* 77.7%

         \[\leadsto x \cdot x - \frac{{z}^{2}}{\color{blue}{\frac{\frac{1}{4}}{y}}} \]

      7. metadata-eval 77.7%

         \[\leadsto x \cdot x - \frac{{z}^{2}}{\frac{\color{blue}{0.25}}{y}} \]

   8. Applied egg-rr 77.7%

      \[\leadsto x \cdot x - \color{blue}{\frac{{z}^{2}}{\frac{0.25}{y}}} \]
   9. Step-by-step derivation

      1. pow2 77.7%

         \[\leadsto x \cdot x - \frac{\color{blue}{z \cdot z}}{\frac{0.25}{y}} \]

      2. clear-num 77.7%

         \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{\frac{0.25}{y}}{z \cdot z}}} \]

      3. associate-/r/ 77.7%

         \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{0.25}{y}} \cdot \left(z \cdot z\right)} \]

      4. clear-num 77.7%

         \[\leadsto x \cdot x - \color{blue}{\frac{y}{0.25}} \cdot \left(z \cdot z\right) \]

      5. div-inv 77.7%

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot \frac{1}{0.25}\right)} \cdot \left(z \cdot z\right) \]

      6. metadata-eval 77.7%

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

      7. associate-*r* 86.4%

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

   10. Applied egg-rr 86.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.6 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 2: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1.6e+21)
   (- (* x x) (* y (* t -4.0)))
   (- (* x x) (* (* z z) (* y 4.0)))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1.6e+21) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (x * x) - ((z * z) * (y * 4.0));
	}
	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 * z) <= 1.6d+21) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (x * x) - ((z * z) * (y * 4.0d0))
    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 * z) <= 1.6e+21) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (x * x) - ((z * z) * (y * 4.0));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1.6e+21:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = (x * x) - ((z * z) * (y * 4.0))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1.6e+21)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(z * z) * Float64(y * 4.0)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1.6e+21)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = (x * x) - ((z * z) * (y * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1.6e+21], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(z * z), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.6e21

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*l* 92.0%

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

   4. Simplified 92.0%

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

   if 1.6e21 < (*.f64 z z)

   1. Initial program 86.8%

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

      1. add-cube-cbrt 86.5%

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

      2. pow3 86.6%

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

      3. pow2 86.6%

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

   3. Applied egg-rr 86.6%

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

   4. Taylor expanded in t around 0 81.5%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot {\color{blue}{\left({\left({z}^{2}\right)}^{0.3333333333333333}\right)}}^{3} \]
   5. Step-by-step derivation

      1. unpow1/3 82.3%

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

   6. Simplified 82.3%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot {\color{blue}{\left(\sqrt[3]{{z}^{2}}\right)}}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 82.5%

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

      2. unpow2 82.5%

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

   8. Applied egg-rr 82.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \end{array} \]

Alternative 3: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+20}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+20)
   (- (* x x) (* y (* t -4.0)))
   (- (* x x) (* z (* z (* y 4.0))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+20) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	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 * z) <= 1d+20) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (x * x) - (z * (z * (y * 4.0d0)))
    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 * z) <= 1e+20) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+20:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = (x * x) - (z * (z * (y * 4.0)))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+20)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+20)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = (x * x) - (z * (z * (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+20], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e20

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*l* 92.0%

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

   4. Simplified 92.0%

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

   if 1e20 < (*.f64 z z)

   1. Initial program 86.8%

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

      1. add-cube-cbrt 86.5%

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

      2. pow3 86.6%

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

      3. pow2 86.6%

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

   3. Applied egg-rr 86.6%

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

   4. Taylor expanded in t around 0 81.5%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot {\color{blue}{\left({\left({z}^{2}\right)}^{0.3333333333333333}\right)}}^{3} \]
   5. Step-by-step derivation

      1. unpow1/3 82.3%

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

   6. Simplified 82.3%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot {\color{blue}{\left(\sqrt[3]{{z}^{2}}\right)}}^{3} \]
   7. Step-by-step derivation

      1. remove-double-div 82.3%

         \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{1}{y \cdot 4}}} \cdot {\left(\sqrt[3]{{z}^{2}}\right)}^{3} \]

      2. rem-cube-cbrt 82.5%

         \[\leadsto x \cdot x - \frac{1}{\frac{1}{y \cdot 4}} \cdot \color{blue}{{z}^{2}} \]

      3. associate-*l/ 82.5%

         \[\leadsto x \cdot x - \color{blue}{\frac{1 \cdot {z}^{2}}{\frac{1}{y \cdot 4}}} \]

      4. *-un-lft-identity 82.5%

         \[\leadsto x \cdot x - \frac{\color{blue}{{z}^{2}}}{\frac{1}{y \cdot 4}} \]

      5. *-commutative 82.5%

         \[\leadsto x \cdot x - \frac{{z}^{2}}{\frac{1}{\color{blue}{4 \cdot y}}} \]

      6. associate-/r* 82.5%

         \[\leadsto x \cdot x - \frac{{z}^{2}}{\color{blue}{\frac{\frac{1}{4}}{y}}} \]

      7. metadata-eval 82.5%

         \[\leadsto x \cdot x - \frac{{z}^{2}}{\frac{\color{blue}{0.25}}{y}} \]

   8. Applied egg-rr 82.5%

      \[\leadsto x \cdot x - \color{blue}{\frac{{z}^{2}}{\frac{0.25}{y}}} \]
   9. Step-by-step derivation

      1. pow2 82.5%

         \[\leadsto x \cdot x - \frac{\color{blue}{z \cdot z}}{\frac{0.25}{y}} \]

      2. clear-num 82.5%

         \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{\frac{0.25}{y}}{z \cdot z}}} \]

      3. associate-/r/ 82.5%

         \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{0.25}{y}} \cdot \left(z \cdot z\right)} \]

      4. clear-num 82.5%

         \[\leadsto x \cdot x - \color{blue}{\frac{y}{0.25}} \cdot \left(z \cdot z\right) \]

      5. div-inv 82.5%

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot \frac{1}{0.25}\right)} \cdot \left(z \cdot z\right) \]

      6. metadata-eval 82.5%

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

      7. associate-*r* 89.0%

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

   10. Applied egg-rr 89.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+20}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 4: 95.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+104}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.45e+104)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (- (* x x) (* z (* z (* y 4.0))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.45e+104) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	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+104) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - (z * (z * (y * 4.0d0)))
    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+104) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if z <= 1.45e+104:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - (z * (z * (y * 4.0)))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.45e+104)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.45e+104)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - (z * (z * (y * 4.0)));
	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+104], 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[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4499999999999999e104

   1. Initial program 95.3%

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

   if 1.4499999999999999e104 < z

   1. Initial program 80.1%

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

      1. add-cube-cbrt 80.0%

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

      2. pow3 80.0%

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

      3. pow2 80.0%

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

   3. Applied egg-rr 80.0%

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

   4. Taylor expanded in t around 0 79.6%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot {\color{blue}{\left({\left({z}^{2}\right)}^{0.3333333333333333}\right)}}^{3} \]
   5. Step-by-step derivation

      1. unpow1/3 80.0%

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

   6. Simplified 80.0%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot {\color{blue}{\left(\sqrt[3]{{z}^{2}}\right)}}^{3} \]
   7. Step-by-step derivation

      1. remove-double-div 80.0%

         \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{1}{y \cdot 4}}} \cdot {\left(\sqrt[3]{{z}^{2}}\right)}^{3} \]

      2. rem-cube-cbrt 80.1%

         \[\leadsto x \cdot x - \frac{1}{\frac{1}{y \cdot 4}} \cdot \color{blue}{{z}^{2}} \]

      3. associate-*l/ 80.2%

         \[\leadsto x \cdot x - \color{blue}{\frac{1 \cdot {z}^{2}}{\frac{1}{y \cdot 4}}} \]

      4. *-un-lft-identity 80.2%

         \[\leadsto x \cdot x - \frac{\color{blue}{{z}^{2}}}{\frac{1}{y \cdot 4}} \]

      5. *-commutative 80.2%

         \[\leadsto x \cdot x - \frac{{z}^{2}}{\frac{1}{\color{blue}{4 \cdot y}}} \]

      6. associate-/r* 80.2%

         \[\leadsto x \cdot x - \frac{{z}^{2}}{\color{blue}{\frac{\frac{1}{4}}{y}}} \]

      7. metadata-eval 80.2%

         \[\leadsto x \cdot x - \frac{{z}^{2}}{\frac{\color{blue}{0.25}}{y}} \]

   8. Applied egg-rr 80.2%

      \[\leadsto x \cdot x - \color{blue}{\frac{{z}^{2}}{\frac{0.25}{y}}} \]
   9. Step-by-step derivation

      1. pow2 80.2%

         \[\leadsto x \cdot x - \frac{\color{blue}{z \cdot z}}{\frac{0.25}{y}} \]

      2. clear-num 80.2%

         \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{\frac{0.25}{y}}{z \cdot z}}} \]

      3. associate-/r/ 80.1%

         \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{0.25}{y}} \cdot \left(z \cdot z\right)} \]

      4. clear-num 80.1%

         \[\leadsto x \cdot x - \color{blue}{\frac{y}{0.25}} \cdot \left(z \cdot z\right) \]

      5. div-inv 80.1%

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot \frac{1}{0.25}\right)} \cdot \left(z \cdot z\right) \]

      6. metadata-eval 80.1%

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

      7. associate-*r* 86.6%

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

   10. Applied egg-rr 86.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+104}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 5: 66.8% accurate, 1.4× speedup

Math

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

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t) :precision binary64 (- (* x x) (* y (* t -4.0))))

C

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

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) - (y * (t * (-4.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

2. Taylor expanded in z around 0 67.8%

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

   1. *-commutative 67.8%

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

   2. *-commutative 67.8%

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

   3. associate-*l* 67.8%

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

4. Simplified 67.8%

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

5. Final simplification 67.8%

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

Alternative 6: 31.5% accurate, 2.6× speedup

Math

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

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t) :precision binary64 (* 4.0 (* t y)))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	return 4.0 * (t * y);
}

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 = 4.0d0 * (t * y)
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	return 4.0 * (t * y);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

2. Taylor expanded in t around inf 26.4%

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

   1. *-commutative 26.4%

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

4. Simplified 26.4%

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

5. Final simplification 26.4%

   \[\leadsto 4 \cdot \left(t \cdot y\right) \]

Alternative 7: 31.4% accurate, 2.6× speedup

Math

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

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t) :precision binary64 (* y (* t 4.0)))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	return y * (t * 4.0);
}

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 = y * (t * 4.0d0)
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	return y * (t * 4.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

2. Taylor expanded in t around inf 26.4%

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

   1. associate-*r* 26.8%

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

   2. *-commutative 26.8%

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

4. Simplified 26.8%

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

5. Final simplification 26.8%

   \[\leadsto y \cdot \left(t \cdot 4\right) \]

Developer target: 89.9% 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 2023306 
(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))))