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

Percentage Accurate: 90.5% → 97.4%

Time: 8.6s

Alternatives: 5

Speedup: 0.6×

• 48.7% 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: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -5 \cdot 10^{+59}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{elif}\;y \cdot 4 \leq 0.2:\\ \;\;\;\;x \cdot x - \left(y \cdot \left(t \cdot -4\right) + z \cdot \left(\left(y \cdot 4\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y 4.0) -5e+59)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (if (<= (* y 4.0) 0.2)
     (- (* x x) (+ (* y (* t -4.0)) (* z (* (* y 4.0) z))))
     (fma x x (* (- (* z z) t) (* y -4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * 4.0) <= -5e+59) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else if ((y * 4.0) <= 0.2) {
		tmp = (x * x) - ((y * (t * -4.0)) + (z * ((y * 4.0) * z)));
	} else {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * 4.0) <= -5e+59)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	elseif (Float64(y * 4.0) <= 0.2)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * Float64(t * -4.0)) + Float64(z * Float64(Float64(y * 4.0) * z))));
	else
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * 4.0), $MachinePrecision], -5e+59], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * 4.0), $MachinePrecision], 0.2], N[(N[(x * x), $MachinePrecision] - N[(N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y 4) < -4.9999999999999997e59

   1. Initial program 100.0%

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

   if -4.9999999999999997e59 < (*.f64 y 4) < 0.20000000000000001

   1. Initial program 89.4%

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

   3. Taylor expanded in z around 0 89.4%

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

   5. Applied egg-rr 49.2%

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

      1. fma-udef 49.2%

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

      2. unpow2 49.2%

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

      3. associate-*r* 49.2%

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

   7. Simplified 49.2%

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

      1. associate-*l* 49.2%

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

      2. unpow-prod-down 43.6%

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

      3. metadata-eval 43.6%

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

      4. sqrt-prod 43.6%

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

      5. *-commutative 43.6%

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

      6. pow2 43.6%

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

      7. add-sqr-sqrt 89.4%

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

      8. *-commutative 89.4%

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

      9. unpow2 89.4%

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

      10. associate-*r* 98.5%

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

      11. *-commutative 98.5%

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

   9. Applied egg-rr 98.5%

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

   if 0.20000000000000001 < (*.f64 y 4)

   1. Initial program 87.7%

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

      1. fma-neg 96.4%

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

      2. distribute-lft-neg-in 96.4%

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

      3. *-commutative 96.4%

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

      4. distribute-rgt-neg-in 96.4%

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

      5. metadata-eval 96.4%

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

   3. Simplified 96.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -5 \cdot 10^{+59}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{elif}\;y \cdot 4 \leq 0.2:\\ \;\;\;\;x \cdot x - \left(y \cdot \left(t \cdot -4\right) + z \cdot \left(\left(y \cdot 4\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* x x) (* (* y 4.0) (- t (* z z))))))
   (if (<= t_1 INFINITY) t_1 (- (* x x) (* y (* t -4.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x * x) - (y * (t * -4.0));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (x * x) - (y * (t * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = (x * x) - (y * (t * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	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)
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = (x * x) - (y * (t * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 96.1%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 50.0%

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

      1. *-commutative 50.0%

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

      2. *-commutative 50.0%

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

      3. associate-*l* 50.0%

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

   5. Simplified 50.0%

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

4. Final simplification 94.0%

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

Alternative 3: 94.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-37}:\\ \;\;\;\;x \cdot x - \left(y \cdot \left(t \cdot -4\right) + z \cdot \left(\left(y \cdot 4\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2e-37)
   (- (* x x) (+ (* y (* t -4.0)) (* z (* (* y 4.0) z))))
   (+ (* x x) (* (* y 4.0) (- t (* z z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2e-37) {
		tmp = (x * x) - ((y * (t * -4.0)) + (z * ((y * 4.0) * z)));
	} else {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	}
	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 (t <= 2d-37) then
        tmp = (x * x) - ((y * (t * (-4.0d0))) + (z * ((y * 4.0d0) * z)))
    else
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2e-37) {
		tmp = (x * x) - ((y * (t * -4.0)) + (z * ((y * 4.0) * z)));
	} else {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 2e-37:
		tmp = (x * x) - ((y * (t * -4.0)) + (z * ((y * 4.0) * z)))
	else:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2e-37)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * Float64(t * -4.0)) + Float64(z * Float64(Float64(y * 4.0) * z))));
	else
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2e-37)
		tmp = (x * x) - ((y * (t * -4.0)) + (z * ((y * 4.0) * z)));
	else
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2e-37], N[(N[(x * x), $MachinePrecision] - N[(N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $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 < 2.00000000000000013e-37

   1. Initial program 90.8%

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

   3. Taylor expanded in z around 0 90.8%

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

   5. Applied egg-rr 43.3%

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

      1. fma-udef 43.2%

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

      2. unpow2 43.2%

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

      3. associate-*r* 43.2%

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

   7. Simplified 43.2%

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

      1. associate-*l* 43.2%

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

      2. unpow-prod-down 39.5%

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

      3. metadata-eval 39.5%

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

      4. sqrt-prod 39.5%

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

      5. *-commutative 39.5%

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

      6. pow2 39.5%

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

      7. add-sqr-sqrt 90.8%

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

      8. *-commutative 90.8%

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

      9. unpow2 90.8%

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

      10. associate-*r* 96.6%

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

      11. *-commutative 96.6%

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

   9. Applied egg-rr 96.6%

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

   if 2.00000000000000013e-37 < t

   1. Initial program 93.6%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-37}:\\ \;\;\;\;x \cdot x - \left(y \cdot \left(t \cdot -4\right) + z \cdot \left(\left(y \cdot 4\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * x) - (y * (t * -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 = (x * x) - (y * (t * (-4.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in z around 0 67.6%

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

   1. *-commutative 67.6%

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

   2. *-commutative 67.6%

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

   3. associate-*l* 67.6%

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

5. Simplified 67.6%

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

6. Final simplification 67.6%

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

Alternative 5: 32.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (y * t) * 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 = (y * t) * 4.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in t around inf 29.0%

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

   1. *-commutative 29.0%

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

5. Simplified 29.0%

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

6. Final simplification 29.0%

   \[\leadsto \left(y \cdot t\right) \cdot 4 \]
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 2024026 
(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))))