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

Percentage Accurate: 90.2% → 91.7%

Time: 9.1s

Alternatives: 6

Speedup: 0.6×

• 49.1% 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: 90.2% 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: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(t \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2e+245)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (fma x x (* y (* t 4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+245) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = fma(x, x, (y * (t * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2e+245)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = fma(x, x, Float64(y * Float64(t * 4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+245], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000009e245

   1. Initial program 94.7%

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

      1. fma-neg 94.7%

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

      2. distribute-lft-neg-in 94.7%

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

      3. *-commutative 94.7%

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

      4. distribute-rgt-neg-in 94.7%

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

      5. metadata-eval 94.7%

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

   3. Simplified 94.7%

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

   if 2.00000000000000009e245 < (*.f64 x x)

   1. Initial program 80.6%

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

   3. Taylor expanded in z around 0 92.4%

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

      1. *-commutative 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-*l* 92.4%

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

   5. Simplified 92.4%

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

      1. fma-neg 96.3%

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

      2. distribute-rgt-neg-in 96.3%

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

      3. distribute-rgt-neg-in 96.3%

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

      4. metadata-eval 96.3%

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

   7. Applied egg-rr 96.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(t \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+245}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(t \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2e+245)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (fma x x (* y (* t 4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+245) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = fma(x, x, (y * (t * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2e+245)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = fma(x, x, Float64(y * Float64(t * 4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+245], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000009e245

   1. Initial program 94.7%

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

   if 2.00000000000000009e245 < (*.f64 x x)

   1. Initial program 80.6%

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

   3. Taylor expanded in z around 0 92.4%

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

      1. *-commutative 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-*l* 92.4%

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

   5. Simplified 92.4%

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

      1. fma-neg 96.3%

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

      2. distribute-rgt-neg-in 96.3%

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

      3. distribute-rgt-neg-in 96.3%

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

      4. metadata-eval 96.3%

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

   7. Applied egg-rr 96.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+245}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(t \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.28 \cdot 10^{-17} \lor \neg \left(z \cdot z \leq 38000000000\right) \land z \cdot z \leq 3.1 \cdot 10^{+182}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z z) 1.28e-17)
         (and (not (<= (* z z) 38000000000.0)) (<= (* z z) 3.1e+182)))
   (- (* x x) (* y (* t -4.0)))
   (* y (* (* z z) -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) <= 1.28e-17) || (!((z * z) <= 38000000000.0) && ((z * z) <= 3.1e+182))) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * z) <= 1.28d-17) .or. (.not. ((z * z) <= 38000000000.0d0)) .and. ((z * z) <= 3.1d+182)) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = y * ((z * z) * (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) <= 1.28e-17) || (!((z * z) <= 38000000000.0) && ((z * z) <= 3.1e+182))) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) <= 1.28e-17) or (not ((z * z) <= 38000000000.0) and ((z * z) <= 3.1e+182)):
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = y * ((z * z) * -4.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * z) <= 1.28e-17) || (!(Float64(z * z) <= 38000000000.0) && (Float64(z * z) <= 3.1e+182)))
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(y * Float64(Float64(z * z) * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) <= 1.28e-17) || (~(((z * z) <= 38000000000.0)) && ((z * z) <= 3.1e+182)))
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = y * ((z * z) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 1.28e-17], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 38000000000.0]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 3.1e+182]]], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.28e-17 or 3.8e10 < (*.f64 z z) < 3.09999999999999996e182

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0 91.5%

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

      1. *-commutative 91.5%

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

      2. *-commutative 91.5%

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

      3. associate-*l* 91.5%

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

   5. Simplified 91.5%

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

   if 1.28e-17 < (*.f64 z z) < 3.8e10 or 3.09999999999999996e182 < (*.f64 z z)

   1. Initial program 77.4%

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

   3. Taylor expanded in z around inf 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l* 71.7%

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

   5. Simplified 71.7%

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

      1. unpow2 71.7%

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

   7. Applied egg-rr 71.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.28 \cdot 10^{-17} \lor \neg \left(z \cdot z \leq 38000000000\right) \land z \cdot z \leq 3.1 \cdot 10^{+182}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5.7 \cdot 10^{-18} \lor \neg \left(z \cdot z \leq 4.8 \cdot 10^{+46}\right) \land z \cdot z \leq 1.35 \cdot 10^{+114}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z z) 5.7e-18)
         (and (not (<= (* z z) 4.8e+46)) (<= (* z z) 1.35e+114)))
   (* t (* y 4.0))
   (* y (* (* z z) -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) <= 5.7e-18) || (!((z * z) <= 4.8e+46) && ((z * z) <= 1.35e+114))) {
		tmp = t * (y * 4.0);
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * z) <= 5.7d-18) .or. (.not. ((z * z) <= 4.8d+46)) .and. ((z * z) <= 1.35d+114)) then
        tmp = t * (y * 4.0d0)
    else
        tmp = y * ((z * z) * (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) <= 5.7e-18) || (!((z * z) <= 4.8e+46) && ((z * z) <= 1.35e+114))) {
		tmp = t * (y * 4.0);
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) <= 5.7e-18) or (not ((z * z) <= 4.8e+46) and ((z * z) <= 1.35e+114)):
		tmp = t * (y * 4.0)
	else:
		tmp = y * ((z * z) * -4.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * z) <= 5.7e-18) || (!(Float64(z * z) <= 4.8e+46) && (Float64(z * z) <= 1.35e+114)))
		tmp = Float64(t * Float64(y * 4.0));
	else
		tmp = Float64(y * Float64(Float64(z * z) * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) <= 5.7e-18) || (~(((z * z) <= 4.8e+46)) && ((z * z) <= 1.35e+114)))
		tmp = t * (y * 4.0);
	else
		tmp = y * ((z * z) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 5.7e-18], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 4.8e+46]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 1.35e+114]]], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.69999999999999971e-18 or 4.80000000000000017e46 < (*.f64 z z) < 1.35e114

   1. Initial program 98.4%

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

   3. Taylor expanded in t around inf 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*r* 50.6%

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

      3. *-commutative 50.6%

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

   5. Simplified 50.6%

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

   if 5.69999999999999971e-18 < (*.f64 z z) < 4.80000000000000017e46 or 1.35e114 < (*.f64 z z)

   1. Initial program 82.1%

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

   3. Taylor expanded in z around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l* 63.7%

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

   5. Simplified 63.7%

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

      1. unpow2 63.7%

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

   7. Applied egg-rr 63.7%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5.7 \cdot 10^{-18} \lor \neg \left(z \cdot z \leq 4.8 \cdot 10^{+46}\right) \land z \cdot z \leq 1.35 \cdot 10^{+114}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8.8 \cdot 10^{+246}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 8.8e+246)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (- (* x x) (* y (* t -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 8.8e+246) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (y * (t * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 8.8d+246) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - (y * (t * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 8.8e+246) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (y * (t * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 8.8e+246:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - (y * (t * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 8.8e+246)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 8.8e+246)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - (y * (t * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 8.8e+246], 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[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 8.79999999999999952e246

   1. Initial program 94.7%

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

   if 8.79999999999999952e246 < (*.f64 x x)

   1. Initial program 80.6%

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

   3. Taylor expanded in z around 0 92.4%

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

      1. *-commutative 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-*l* 92.4%

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

   5. Simplified 92.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8.8 \cdot 10^{+246}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t * (y * 4.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in t around inf 31.7%

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

   1. *-commutative 31.7%

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

   2. associate-*r* 31.7%

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

   3. *-commutative 31.7%

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

5. Simplified 31.7%

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

6. Final simplification 31.7%

   \[\leadsto t \cdot \left(y \cdot 4\right) \]
7. Add Preprocessing

Developer target: 90.3% 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 2024017 
(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))))