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

Percentage Accurate: 91.1% → 94.8%

Time: 10.3s

Alternatives: 6

Speedup: 0.7×

• 49.9% 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: 91.1% 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: 94.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot \frac{z}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+302)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (* -4.0 (* t (* (* z y) (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+302) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = -4.0 * (t * ((z * y) * (z / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+302)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(Float64(z * y) * Float64(z / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+302], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * N[(N[(z * y), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e302

   1. Initial program 99.4%

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

      1. fma-neg 100.0%

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

      2. distribute-lft-neg-in 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   if 5e302 < (*.f64 z z)

   1. Initial program 67.0%

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

   3. Taylor expanded in t around inf 67.0%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(t \cdot \left(\frac{{z}^{2}}{t} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 67.0%

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

      2. associate-/l* 67.0%

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

      3. fma-neg 67.0%

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

      4. metadata-eval 67.0%

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

   5. Simplified 67.0%

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

   6. Taylor expanded in x around 0 72.2%

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \left(\frac{{z}^{2}}{t} - 1\right)\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 72.2%

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

      2. associate-*r/ 81.2%

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

      3. fma-neg 81.2%

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

      4. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in z around inf 72.2%

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

       1. associate-/l* 72.2%

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

       2. unpow2 72.2%

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

       3. associate-*r/ 81.2%

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

       4. associate-*r* 86.0%

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

   11. Applied egg-rr 86.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot \frac{z}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{+151}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot \frac{z}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 9.5e+151)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* -4.0 (* t (* (* z y) (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 9.5e+151) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = -4.0 * (t * ((z * y) * (z / t)));
	}
	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 <= 9.5d+151) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (-4.0d0) * (t * ((z * y) * (z / t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 9.5e+151) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = -4.0 * (t * ((z * y) * (z / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 9.5e+151:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = -4.0 * (t * ((z * y) * (z / t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 9.5e+151)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(Float64(z * y) * Float64(z / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 9.5e+151)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = -4.0 * (t * ((z * y) * (z / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 9.5e+151], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * N[(N[(z * y), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.5000000000000001e151

   1. Initial program 96.9%

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

   if 9.5000000000000001e151 < z

   1. Initial program 66.2%

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

   3. Taylor expanded in t around inf 66.2%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(t \cdot \left(\frac{{z}^{2}}{t} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 66.2%

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

      2. associate-/l* 66.2%

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

      3. fma-neg 66.2%

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

      4. metadata-eval 66.2%

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

   5. Simplified 66.2%

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

   6. Taylor expanded in x around 0 73.7%

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \left(\frac{{z}^{2}}{t} - 1\right)\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 73.7%

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

      2. associate-*r/ 80.7%

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

      3. fma-neg 80.7%

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

      4. metadata-eval 80.7%

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

   8. Simplified 80.7%

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

   9. Taylor expanded in z around inf 73.7%

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

       1. associate-/l* 73.7%

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

       2. unpow2 73.7%

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

       3. associate-*r/ 80.7%

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

       4. associate-*r* 83.1%

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

   11. Applied egg-rr 83.1%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{+151}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot \frac{z}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+38}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \left(\frac{z}{\frac{t}{z}} + -1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 5.8e+38)
   (- (* x x) (* y (* t -4.0)))
   (* -4.0 (* t (* y (+ (/ z (/ t z)) -1.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.8e+38) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = -4.0 * (t * (y * ((z / (t / z)) + -1.0)));
	}
	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 <= 5.8d+38) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (-4.0d0) * (t * (y * ((z / (t / z)) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.8e+38) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = -4.0 * (t * (y * ((z / (t / z)) + -1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 5.8e+38:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = -4.0 * (t * (y * ((z / (t / z)) + -1.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 5.8e+38)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(y * Float64(Float64(z / Float64(t / z)) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 5.8e+38)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = -4.0 * (t * (y * ((z / (t / z)) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 5.8e+38], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * N[(y * N[(N[(z / N[(t / z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.80000000000000013e38

   1. Initial program 96.5%

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

   3. Taylor expanded in z around 0 80.2%

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

      1. *-commutative 80.2%

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

      2. *-commutative 80.2%

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

      3. associate-*l* 80.2%

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

   5. Simplified 80.2%

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

   if 5.80000000000000013e38 < z

   1. Initial program 79.8%

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

   3. Taylor expanded in t around inf 74.1%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(t \cdot \left(\frac{{z}^{2}}{t} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 74.1%

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

      2. associate-/l* 74.1%

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

      3. fma-neg 74.1%

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

      4. metadata-eval 74.1%

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

   5. Simplified 74.1%

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

   6. Taylor expanded in x around 0 70.2%

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \left(\frac{{z}^{2}}{t} - 1\right)\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 70.2%

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

      2. *-un-lft-identity 70.2%

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

      3. times-frac 74.4%

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

   8. Applied egg-rr 74.4%

      \[\leadsto -4 \cdot \left(t \cdot \left(y \cdot \left(\color{blue}{\frac{z}{1} \cdot \frac{z}{t}} - 1\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-rgt-identity 74.4%

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

      2. clear-num 74.4%

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

      3. un-div-inv 74.4%

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

   10. Applied egg-rr 74.4%

       \[\leadsto -4 \cdot \left(t \cdot \left(y \cdot \left(\color{blue}{\frac{z}{\frac{t}{z}}} - 1\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+38}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \left(\frac{z}{\frac{t}{z}} + -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+83}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot \frac{z}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.4e+83)
   (- (* x x) (* y (* t -4.0)))
   (* -4.0 (* t (* (* z y) (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.4e+83) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = -4.0 * (t * ((z * y) * (z / t)));
	}
	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 <= 2.4d+83) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (-4.0d0) * (t * ((z * y) * (z / t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.4e+83) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = -4.0 * (t * ((z * y) * (z / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 2.4e+83:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = -4.0 * (t * ((z * y) * (z / t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2.4e+83)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(Float64(z * y) * Float64(z / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2.4e+83)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = -4.0 * (t * ((z * y) * (z / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 2.4e+83], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * N[(N[(z * y), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.39999999999999991e83

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0 78.0%

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

      1. *-commutative 78.0%

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

      2. *-commutative 78.0%

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

      3. associate-*l* 78.0%

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

   5. Simplified 78.0%

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

   if 2.39999999999999991e83 < z

   1. Initial program 75.9%

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

   3. Taylor expanded in t around inf 70.7%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(t \cdot \left(\frac{{z}^{2}}{t} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 70.7%

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

      2. associate-/l* 70.7%

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

      3. fma-neg 70.7%

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

      4. metadata-eval 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in x around 0 74.3%

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \left(\frac{{z}^{2}}{t} - 1\right)\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 74.3%

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

      2. associate-*r/ 79.4%

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

      3. fma-neg 79.4%

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

      4. metadata-eval 79.4%

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

   8. Simplified 79.4%

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

   9. Taylor expanded in z around inf 69.0%

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

       1. associate-/l* 69.0%

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

       2. unpow2 69.0%

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

       3. associate-*r/ 74.0%

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

       4. associate-*r* 75.7%

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

   11. Applied egg-rr 75.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+83}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot \frac{z}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% 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 92.1%

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

3. Taylor expanded in z around 0 66.7%

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

   1. *-commutative 66.7%

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

   2. *-commutative 66.7%

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

   3. associate-*l* 66.7%

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

5. Simplified 66.7%

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

Alternative 6: 31.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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 = 4.0d0 * (t * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in t around inf 37.1%

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

   1. *-commutative 37.1%

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

5. Simplified 37.1%

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

6. Final simplification 37.1%

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

Developer target: 91.1% 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 2024108 
(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))))