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

Percentage Accurate: 91.3% → 94.4%

Time: 9.8s

Alternatives: 7

Speedup: 0.7×

• 49.7% 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: 91.3% 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: 94.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 4e+203) (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 <= 4e+203) {
		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 <= 4e+203)
		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, 4e+203], 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 < 4e203

   1. Initial program 91.7%

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

      1. fma-neg 93.9%

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

      2. distribute-lft-neg-in 93.9%

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

      3. *-commutative 93.9%

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

      4. distribute-rgt-neg-in 93.9%

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

      5. metadata-eval 93.9%

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

   3. Simplified 93.9%

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

   if 4e203 < x

   1. Initial program 94.7%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. pow2 100.0%

         \[\leadsto \color{blue}{x \cdot x} \]

   5. Applied egg-rr 100.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_1 := 4 \cdot \left(t \cdot y\right)\\ \mathbf{if}\;z \leq 6 \cdot 10^{-171}:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-32}:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (let* ((t_1 (* 4.0 (* t y))))
   (if (<= z 6e-171)
     (* x_m x_m)
     (if (<= z 5.8e-71)
       t_1
       (if (<= z 6.6e-32)
         (* x_m x_m)
         (if (<= z 2.9e+15) t_1 (* z (* z (* y -4.0)))))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double t_1 = 4.0 * (t * y);
	double tmp;
	if (z <= 6e-171) {
		tmp = x_m * x_m;
	} else if (z <= 5.8e-71) {
		tmp = t_1;
	} else if (z <= 6.6e-32) {
		tmp = x_m * x_m;
	} else if (z <= 2.9e+15) {
		tmp = t_1;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = 4.0d0 * (t * y)
    if (z <= 6d-171) then
        tmp = x_m * x_m
    else if (z <= 5.8d-71) then
        tmp = t_1
    else if (z <= 6.6d-32) then
        tmp = x_m * x_m
    else if (z <= 2.9d+15) then
        tmp = t_1
    else
        tmp = z * (z * (y * (-4.0d0)))
    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 t_1 = 4.0 * (t * y);
	double tmp;
	if (z <= 6e-171) {
		tmp = x_m * x_m;
	} else if (z <= 5.8e-71) {
		tmp = t_1;
	} else if (z <= 6.6e-32) {
		tmp = x_m * x_m;
	} else if (z <= 2.9e+15) {
		tmp = t_1;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	t_1 = 4.0 * (t * y)
	tmp = 0
	if z <= 6e-171:
		tmp = x_m * x_m
	elif z <= 5.8e-71:
		tmp = t_1
	elif z <= 6.6e-32:
		tmp = x_m * x_m
	elif z <= 2.9e+15:
		tmp = t_1
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	t_1 = Float64(4.0 * Float64(t * y))
	tmp = 0.0
	if (z <= 6e-171)
		tmp = Float64(x_m * x_m);
	elseif (z <= 5.8e-71)
		tmp = t_1;
	elseif (z <= 6.6e-32)
		tmp = Float64(x_m * x_m);
	elseif (z <= 2.9e+15)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	t_1 = 4.0 * (t * y);
	tmp = 0.0;
	if (z <= 6e-171)
		tmp = x_m * x_m;
	elseif (z <= 5.8e-71)
		tmp = t_1;
	elseif (z <= 6.6e-32)
		tmp = x_m * x_m;
	elseif (z <= 2.9e+15)
		tmp = t_1;
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 6e-171], N[(x$95$m * x$95$m), $MachinePrecision], If[LessEqual[z, 5.8e-71], t$95$1, If[LessEqual[z, 6.6e-32], N[(x$95$m * x$95$m), $MachinePrecision], If[LessEqual[z, 2.9e+15], t$95$1, N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.9999999999999999e-171 or 5.7999999999999997e-71 < z < 6.60000000000000051e-32

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf 44.1%

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. pow2 44.1%

         \[\leadsto \color{blue}{x \cdot x} \]

   5. Applied egg-rr 44.1%

      \[\leadsto \color{blue}{x \cdot x} \]

   if 5.9999999999999999e-171 < z < 5.7999999999999997e-71 or 6.60000000000000051e-32 < z < 2.9e15

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf 62.4%

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

      1. *-commutative 62.4%

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

   5. Simplified 62.4%

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

   if 2.9e15 < z

   1. Initial program 80.1%

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

   3. Taylor expanded in z around inf 65.4%

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

      1. associate-*r* 65.4%

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

   5. Simplified 65.4%

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

      1. add-cube-cbrt 65.2%

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

      2. pow3 65.1%

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

      3. associate-*l* 65.1%

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

   7. Applied egg-rr 65.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{-4 \cdot \left(y \cdot {z}^{2}\right)}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 65.4%

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

      2. associate-*r* 65.4%

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

      3. metadata-eval 65.4%

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

      4. distribute-lft-neg-in 65.4%

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

      5. *-commutative 65.4%

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

      6. unpow2 65.4%

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

      7. associate-*r* 63.6%

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

      8. distribute-rgt-neg-in 63.6%

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

      9. metadata-eval 63.6%

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

   9. Applied egg-rr 63.6%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{-171}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-71}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-32}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+15}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+141}:\\ \;\;\;\;x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 1.7e+141)
   (+ (* x_m x_m) (* (* y 4.0) (- t (* z z))))
   (* z (* z (* y -4.0)))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.7e+141) {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	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 (z <= 1.7d+141) then
        tmp = (x_m * x_m) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = z * (z * (y * (-4.0d0)))
    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 (z <= 1.7e+141) {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if z <= 1.7e+141:
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 1.7e+141)
		tmp = Float64(Float64(x_m * x_m) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if (z <= 1.7e+141)
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 1.7e+141], 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[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6999999999999999e141

   1. Initial program 95.2%

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

   if 1.6999999999999999e141 < z

   1. Initial program 69.6%

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

   3. Taylor expanded in z around inf 75.9%

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

      1. associate-*r* 75.9%

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

   5. Simplified 75.9%

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

      1. add-cube-cbrt 75.8%

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

      2. pow3 75.8%

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

      3. associate-*l* 75.8%

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

   7. Applied egg-rr 75.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{-4 \cdot \left(y \cdot {z}^{2}\right)}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 75.9%

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

      2. associate-*r* 75.9%

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

      3. metadata-eval 75.9%

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

      4. distribute-lft-neg-in 75.9%

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

      5. *-commutative 75.9%

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

      6. unpow2 75.9%

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

      7. associate-*r* 73.1%

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

      8. distribute-rgt-neg-in 73.1%

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

      9. metadata-eval 73.1%

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

   9. Applied egg-rr 73.1%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+141}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{+15}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 9.5e+15) (- (* x_m x_m) (* y (* t -4.0))) (* z (* z (* y -4.0)))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 9.5e+15) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	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 (z <= 9.5d+15) then
        tmp = (x_m * x_m) - (y * (t * (-4.0d0)))
    else
        tmp = z * (z * (y * (-4.0d0)))
    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 (z <= 9.5e+15) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if z <= 9.5e+15:
		tmp = (x_m * x_m) - (y * (t * -4.0))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 9.5e+15)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if (z <= 9.5e+15)
		tmp = (x_m * x_m) - (y * (t * -4.0));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 9.5e+15], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.5e15

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0 71.5%

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

      1. *-commutative 71.5%

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

      2. *-commutative 71.5%

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

      3. associate-*l* 71.5%

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

   5. Simplified 71.5%

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

   if 9.5e15 < z

   1. Initial program 80.1%

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

   3. Taylor expanded in z around inf 65.4%

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

      1. associate-*r* 65.4%

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

   5. Simplified 65.4%

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

      1. add-cube-cbrt 65.2%

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

      2. pow3 65.1%

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

      3. associate-*l* 65.1%

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

   7. Applied egg-rr 65.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{-4 \cdot \left(y \cdot {z}^{2}\right)}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 65.4%

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

      2. associate-*r* 65.4%

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

      3. metadata-eval 65.4%

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

      4. distribute-lft-neg-in 65.4%

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

      5. *-commutative 65.4%

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

      6. unpow2 65.4%

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

      7. associate-*r* 63.6%

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

      8. distribute-rgt-neg-in 63.6%

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

      9. metadata-eval 63.6%

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

   9. Applied egg-rr 63.6%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{+15}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 5.5 \cdot 10^{-78}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* x_m x_m) 5.5e-78) (* 4.0 (* t y)) (* 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) <= 5.5e-78) {
		tmp = 4.0 * (t * y);
	} 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) <= 5.5d-78) then
        tmp = 4.0d0 * (t * y)
    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) <= 5.5e-78) {
		tmp = 4.0 * (t * y);
	} 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) <= 5.5e-78:
		tmp = 4.0 * (t * y)
	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) <= 5.5e-78)
		tmp = Float64(4.0 * Float64(t * y));
	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) <= 5.5e-78)
		tmp = 4.0 * (t * y);
	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], 5.5e-78], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.50000000000000017e-78

   1. Initial program 94.5%

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

   3. Taylor expanded in t around inf 49.6%

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

      1. *-commutative 49.6%

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

   5. Simplified 49.6%

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

   if 5.50000000000000017e-78 < (*.f64 x x)

   1. Initial program 89.6%

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

   3. Taylor expanded in x around inf 70.7%

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. pow2 70.8%

         \[\leadsto \color{blue}{x \cdot x} \]

   5. Applied egg-rr 70.8%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.5 \cdot 10^{-78}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* x_m x_m) 1.25e-80) (* 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) <= 1.25e-80) {
		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) <= 1.25d-80) 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) <= 1.25e-80) {
		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) <= 1.25e-80:
		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) <= 1.25e-80)
		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) <= 1.25e-80)
		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], 1.25e-80], 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) < 1.25e-80

   1. Initial program 94.5%

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

   3. Taylor expanded in t around inf 49.6%

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

      1. associate-*r* 49.6%

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

      2. *-commutative 49.6%

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

   5. Simplified 49.6%

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

   if 1.25e-80 < (*.f64 x x)

   1. Initial program 89.6%

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

   3. Taylor expanded in x around inf 70.7%

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. pow2 70.8%

         \[\leadsto \color{blue}{x \cdot x} \]

   5. Applied egg-rr 70.8%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot x\_m \end{array} \]

FPCore

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

C

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

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
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;
}

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := N[(x$95$m * x$95$m), $MachinePrecision]

Derivation

1. Initial program 92.0%

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

3. Taylor expanded in x around inf 41.5%

   \[\leadsto \color{blue}{{x}^{2}} \]
4. Step-by-step derivation

   1. pow2 41.5%

      \[\leadsto \color{blue}{x \cdot x} \]

5. Applied egg-rr 41.5%

   \[\leadsto \color{blue}{x \cdot x} \]

6. Final simplification 41.5%

   \[\leadsto x \cdot x \]
7. Add Preprocessing

Developer target: 91.4% 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 2024040 
(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))))