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

Percentage Accurate: 90.4% → 93.7%

Time: 10.3s

Alternatives: 6

Speedup: 0.6×

• 49.8% 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: 90.4% 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: 93.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 1.5e+239) (fma x_m x_m (* (- (* z z) t) (* y -4.0))) (* x_m x_m)))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 1.5e+239) {
		tmp = fma(x_m, x_m, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = x_m * x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 1.5e+239)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(x_m * x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[x$95$m, 1.5e+239], N[(x$95$m * x$95$m + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4999999999999999e239

   1. Initial program 93.9%

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

      1. fma-neg 96.0%

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

      2. distribute-lft-neg-in 96.0%

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

      3. *-commutative 96.0%

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

      4. distribute-rgt-neg-in 96.0%

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

      5. metadata-eval 96.0%

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

   3. Simplified 96.0%

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

   if 1.4999999999999999e239 < x

   1. Initial program 66.7%

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

   3. Taylor expanded in y around 0 66.7%

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

   5. Simplified 100.0%

      \[\leadsto x \cdot x - \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{+188}:\\ \;\;\;\;x\_m \cdot x\_m - \left(-4 \cdot \left(t \cdot y\right) + z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= t 9.2e+188)
   (- (* x_m x_m) (+ (* -4.0 (* t y)) (* z (* z (* y 4.0)))))
   (+ (* x_m x_m) (* (* y 4.0) (- t (* z z))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 9.2e+188) {
		tmp = (x_m * x_m) - ((-4.0 * (t * y)) + (z * (z * (y * 4.0))));
	} else {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z, t)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 9.2d+188) then
        tmp = (x_m * x_m) - (((-4.0d0) * (t * y)) + (z * (z * (y * 4.0d0))))
    else
        tmp = (x_m * x_m) + ((y * 4.0d0) * (t - (z * z)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 9.2e+188) {
		tmp = (x_m * x_m) - ((-4.0 * (t * y)) + (z * (z * (y * 4.0))));
	} else {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if t <= 9.2e+188:
		tmp = (x_m * x_m) - ((-4.0 * (t * y)) + (z * (z * (y * 4.0))))
	else:
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (t <= 9.2e+188)
		tmp = Float64(Float64(x_m * x_m) - Float64(Float64(-4.0 * Float64(t * y)) + Float64(z * Float64(z * Float64(y * 4.0)))));
	else
		tmp = Float64(Float64(x_m * x_m) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if (t <= 9.2e+188)
		tmp = (x_m * x_m) - ((-4.0 * (t * y)) + (z * (z * (y * 4.0))));
	else
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[t, 9.2e+188], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.20000000000000046e188

   1. Initial program 92.9%

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

   3. Taylor expanded in z around 0 92.5%

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

      1. +-commutative 92.5%

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

      2. add-sqr-sqrt 59.3%

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

      3. *-commutative 59.3%

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

      4. fma-define 59.3%

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

   5. Applied egg-rr 49.5%

      \[\leadsto x \cdot x - \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sqrt{y}\right) \cdot z, \left(2 \cdot \sqrt{y}\right) \cdot z, \left(-4 \cdot y\right) \cdot t\right)} \]
   6. Step-by-step derivation

      1. fma-undefine 49.5%

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

      2. unpow2 49.5%

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

      3. *-commutative 49.5%

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

      4. associate-*l* 49.5%

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

      5. *-commutative 49.5%

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

   7. Simplified 49.5%

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

      1. unpow-prod-down 45.6%

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

      2. unpow2 45.6%

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

      3. associate-*l* 49.5%

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

      4. *-commutative 49.5%

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

      5. unpow-prod-down 49.5%

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

      6. pow2 49.5%

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

      7. metadata-eval 49.5%

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

      8. add-sqr-sqrt 96.4%

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

   9. Applied egg-rr 96.4%

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

   if 9.20000000000000046e188 < t

   1. Initial program 90.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{+188}:\\ \;\;\;\;x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 3.9 \cdot 10^{+300}:\\ \;\;\;\;x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* x_m x_m) 3.9e+300)
   (+ (* x_m x_m) (* (* y 4.0) (- t (* z z))))
   (* x_m x_m)))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * x_m) <= 3.9e+300) {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x_m * x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z, t)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * x_m) <= 3.9d+300) then
        tmp = (x_m * x_m) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = x_m * x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * x_m) <= 3.9e+300) {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x_m * x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if (x_m * x_m) <= 3.9e+300:
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = x_m * x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 3.9e+300)
		tmp = Float64(Float64(x_m * x_m) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(x_m * x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * x_m) <= 3.9e+300)
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = x_m * x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 3.9e+300], N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.8999999999999999e300

   1. Initial program 95.9%

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

   if 3.8999999999999999e300 < (*.f64 x x)

   1. Initial program 82.8%

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

   3. Taylor expanded in y around 0 82.8%

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

   5. Simplified 92.2%

      \[\leadsto x \cdot x - \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.9 \cdot 10^{+300}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.1% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* x_m x_m) 5e-10) (* y (* t 4.0)) (* x_m x_m)))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * x_m) <= 5e-10) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x_m * x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z, t)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * x_m) <= 5d-10) then
        tmp = y * (t * 4.0d0)
    else
        tmp = x_m * x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * x_m) <= 5e-10) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x_m * x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if (x_m * x_m) <= 5e-10:
		tmp = y * (t * 4.0)
	else:
		tmp = x_m * x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 5e-10)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x_m * x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * x_m) <= 5e-10)
		tmp = y * (t * 4.0);
	else
		tmp = x_m * x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 5e-10], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000031e-10

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf 47.4%

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

      1. associate-*r* 47.4%

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

   5. Simplified 47.4%

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

   if 5.00000000000000031e-10 < (*.f64 x x)

   1. Initial program 88.5%

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

   3. Taylor expanded in y around 0 88.5%

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

   5. Simplified 75.3%

      \[\leadsto x \cdot x - \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

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

Alternative 5: 66.7% accurate, 1.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t) :precision binary64 (- (* x_m x_m) (* -4.0 (* t y))))

C

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

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	return (x_m * x_m) - (-4.0 * (t * y))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(-4.0 * N[(t * y), $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. Add Preprocessing

3. Taylor expanded in z around 0 66.6%

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

   1. *-commutative 66.6%

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

5. Simplified 66.6%

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

6. Final simplification 66.6%

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

Alternative 6: 31.2% accurate, 2.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t) :precision binary64 (* y (* t 4.0)))

C

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

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	return y * (t * 4.0)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, 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. Add Preprocessing

3. Taylor expanded in t around inf 32.7%

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

   1. associate-*r* 32.7%

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

5. Simplified 32.7%

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

6. Final simplification 32.7%

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

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 2024079 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

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

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