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

Percentage Accurate: 90.5% → 95.5%

Time: 9.3s

Alternatives: 8

Speedup: 0.9×

• 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: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.26 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.26e+169)
   (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 <= 1.26e+169) {
		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 <= 1.26e+169)
		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, 1.26e+169], 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 < 1.2599999999999999e169

   1. Initial program 92.4%

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

      1. fma-neg 95.6%

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

      2. distribute-lft-neg-in 95.6%

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

      3. *-commutative 95.6%

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

      4. distribute-rgt-neg-in 95.6%

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

      5. metadata-eval 95.6%

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

   3. Simplified 95.6%

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

   if 1.2599999999999999e169 < z

   1. Initial program 64.9%

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

      1. fma-neg 64.9%

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

   3. Applied egg-rr 64.9%

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

   5. Applied egg-rr 64.9%

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{\mathsf{fma}\left(z, z, t\right)}}} \]

   6. Taylor expanded in z around inf 64.9%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 64.9%

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

      2. /-rgt-identity 64.9%

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

      3. pow2 64.9%

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

      4. associate-*r* 90.8%

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

   8. Applied egg-rr 90.8%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.26 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 2: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 30000000000000 \lor \neg \left(z \cdot z \leq 3.6 \cdot 10^{+41}\right) \land z \cdot z \leq 1.3 \cdot 10^{+131}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z z) 30000000000000.0)
         (and (not (<= (* z z) 3.6e+41)) (<= (* z z) 1.3e+131)))
   (- (* 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) <= 30000000000000.0) || (!((z * z) <= 3.6e+41) && ((z * z) <= 1.3e+131))) {
		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) <= 30000000000000.0d0) .or. (.not. ((z * z) <= 3.6d+41)) .and. ((z * z) <= 1.3d+131)) 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) <= 30000000000000.0) || (!((z * z) <= 3.6e+41) && ((z * z) <= 1.3e+131))) {
		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) <= 30000000000000.0) or (not ((z * z) <= 3.6e+41) and ((z * z) <= 1.3e+131)):
		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) <= 30000000000000.0) || (!(Float64(z * z) <= 3.6e+41) && (Float64(z * z) <= 1.3e+131)))
		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) <= 30000000000000.0) || (~(((z * z) <= 3.6e+41)) && ((z * z) <= 1.3e+131)))
		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[Or[LessEqual[N[(z * z), $MachinePrecision], 30000000000000.0], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 3.6e+41]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 1.3e+131]]], 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) < 3e13 or 3.60000000000000025e41 < (*.f64 z z) < 1.3e131

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 93.6%

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

      1. *-commutative 93.6%

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

      2. *-commutative 93.6%

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

      3. associate-*l* 93.6%

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

   4. Simplified 93.6%

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

   if 3e13 < (*.f64 z z) < 3.60000000000000025e41 or 1.3e131 < (*.f64 z z)

   1. Initial program 73.9%

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

      1. add-cbrt-cube 61.7%

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

      2. pow3 61.7%

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

      3. pow2 61.7%

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

   3. Applied egg-rr 61.7%

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

   4. Taylor expanded in z around inf 60.9%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \sqrt[3]{\color{blue}{{z}^{6}}} \]
   5. Step-by-step derivation

      1. pow1/3 60.7%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{{\left({z}^{6}\right)}^{0.3333333333333333}} \]

      2. pow-pow 71.7%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{{z}^{\left(6 \cdot 0.3333333333333333\right)}} \]

      3. metadata-eval 71.7%

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

      4. unpow2 71.7%

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

   6. Applied egg-rr 71.7%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 30000000000000 \lor \neg \left(z \cdot z \leq 3.6 \cdot 10^{+41}\right) \land z \cdot z \leq 1.3 \cdot 10^{+131}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \end{array} \]

Alternative 3: 88.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-7} \lor \neg \left(z \cdot z \leq 10^{+41}\right) \land z \cdot z \leq 2 \cdot 10^{+131}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z z) 1e-7) (and (not (<= (* z z) 1e+41)) (<= (* z z) 2e+131)))
   (- (* 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-7) || (!((z * z) <= 1e+41) && ((z * z) <= 2e+131))) {
		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-7) .or. (.not. ((z * z) <= 1d+41)) .and. ((z * z) <= 2d+131)) 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-7) || (!((z * z) <= 1e+41) && ((z * z) <= 2e+131))) {
		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-7) or (not ((z * z) <= 1e+41) and ((z * z) <= 2e+131)):
		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-7) || (!(Float64(z * z) <= 1e+41) && (Float64(z * z) <= 2e+131)))
		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-7) || (~(((z * z) <= 1e+41)) && ((z * z) <= 2e+131)))
		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[Or[LessEqual[N[(z * z), $MachinePrecision], 1e-7], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 1e+41]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 2e+131]]], 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) < 9.9999999999999995e-8 or 1.00000000000000001e41 < (*.f64 z z) < 1.9999999999999998e131

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 93.6%

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

      1. *-commutative 93.6%

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

      2. *-commutative 93.6%

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

      3. associate-*l* 93.6%

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

   4. Simplified 93.6%

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

   if 9.9999999999999995e-8 < (*.f64 z z) < 1.00000000000000001e41 or 1.9999999999999998e131 < (*.f64 z z)

   1. Initial program 73.9%

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

      1. fma-neg 73.9%

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

   3. Applied egg-rr 73.9%

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

   5. Applied egg-rr 70.6%

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{\mathsf{fma}\left(z, z, t\right)}}} \]

   6. Taylor expanded in z around inf 71.7%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 71.7%

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

      2. /-rgt-identity 71.7%

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

      3. pow2 71.7%

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

      4. associate-*r* 84.8%

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

   8. Applied egg-rr 84.8%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-7} \lor \neg \left(z \cdot z \leq 10^{+41}\right) \land z \cdot z \leq 2 \cdot 10^{+131}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 4: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+115}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.2e+115)
   (+ (* 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 <= 1.2e+115) {
		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 <= 1.2d+115) 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 <= 1.2e+115) {
		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 <= 1.2e+115:
		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 <= 1.2e+115)
		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 <= 1.2e+115)
		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, 1.2e+115], 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 < 1.2e115

   1. Initial program 94.2%

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

   if 1.2e115 < z

   1. Initial program 66.9%

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

      1. fma-neg 66.9%

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

   3. Applied egg-rr 66.9%

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

   5. Applied egg-rr 66.9%

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{\mathsf{fma}\left(z, z, t\right)}}} \]

   6. Taylor expanded in z around inf 66.9%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 66.9%

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

      2. /-rgt-identity 66.9%

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

      3. pow2 66.9%

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

      4. associate-*r* 85.9%

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

   8. Applied egg-rr 85.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+115}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 5: 66.5% 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 88.9%

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

2. Taylor expanded in z around 0 65.1%

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

   1. *-commutative 65.1%

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

   2. *-commutative 65.1%

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

   3. associate-*l* 65.1%

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

4. Simplified 65.1%

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

5. Final simplification 65.1%

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

Alternative 6: 31.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5.7 \cdot 10^{+137}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 5.7e+137) (* y (* t 4.0)) (* y (* t -4.0))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.7e+137) {
		tmp = y * (t * 4.0);
	} else {
		tmp = y * (t * -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 <= 5.7d+137) then
        tmp = y * (t * 4.0d0)
    else
        tmp = y * (t * (-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 <= 5.7e+137) {
		tmp = y * (t * 4.0);
	} else {
		tmp = y * (t * -4.0);
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if z <= 5.7e+137:
		tmp = y * (t * 4.0)
	else:
		tmp = y * (t * -4.0)
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 5.7e+137)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(y * Float64(t * -4.0));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 5.7e+137)
		tmp = y * (t * 4.0);
	else
		tmp = y * (t * -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, 5.7e+137], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.6999999999999999e137

   1. Initial program 94.0%

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

   2. Taylor expanded in z around 0 73.9%

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

      1. *-commutative 73.9%

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

      2. *-commutative 73.9%

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

      3. associate-*l* 73.9%

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

   4. Simplified 73.9%

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

   5. Taylor expanded in x around 0 44.3%

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

      1. *-commutative 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*r* 44.3%

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

      4. *-commutative 44.3%

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

   7. Simplified 44.3%

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

   if 5.6999999999999999e137 < z

   1. Initial program 63.0%

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

      1. fma-neg 63.0%

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

   3. Applied egg-rr 63.0%

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

   5. Applied egg-rr 63.0%

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{\mathsf{fma}\left(z, z, t\right)}}} \]

   6. Taylor expanded in t around inf 16.0%

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

      1. associate-*r* 16.0%

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

      2. *-commutative 16.0%

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

      3. *-commutative 16.0%

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

   8. Simplified 16.0%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.7 \cdot 10^{+137}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot -4\right)\\ \end{array} \]

Alternative 7: 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 88.9%

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

2. Taylor expanded in t around inf 37.8%

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

   1. *-commutative 37.8%

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

4. Simplified 37.8%

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

5. Final simplification 37.8%

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

Alternative 8: 31.6% 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 88.9%

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

2. Taylor expanded in z around 0 65.1%

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

   1. *-commutative 65.1%

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

   2. *-commutative 65.1%

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

   3. associate-*l* 65.1%

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

4. Simplified 65.1%

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

5. Taylor expanded in x around 0 37.8%

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

   1. *-commutative 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*r* 37.8%

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

   4. *-commutative 37.8%

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

7. Simplified 37.8%

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

8. Final simplification 37.8%

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

Developer target: 90.5% 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 2023311 
(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))))