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

Percentage Accurate: 90.0% → 93.5%

Time: 11.4s

Alternatives: 7

Speedup: 0.7×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.0% 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: 93.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 5e+195) (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 <= 5e+195) {
		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 <= 5e+195)
		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, 5e+195], 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 < 4.9999999999999998e195

   1. Initial program 92.0%

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

      1. fma-neg 94.7%

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

      2. distribute-lft-neg-in 94.7%

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

      3. *-commutative 94.7%

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

      4. distribute-rgt-neg-in 94.7%

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

      5. metadata-eval 94.7%

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

   3. Simplified 94.7%

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

   if 4.9999999999999998e195 < x

   1. Initial program 72.7%

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

   3. Taylor expanded in y around 0 72.7%

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

   5. Simplified 90.9%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 90.9%

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

   7. Applied egg-rr 90.9%

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

Alternative 2: 57.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_1 := \left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \mathbf{if}\;x\_m \leq 2.1 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x\_m \leq 8 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x\_m \leq 3.4 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (let* ((t_1 (* (* z z) (* y -4.0))))
   (if (<= x_m 2.1e-242)
     t_1
     (if (<= x_m 8e-160)
       (* y (* t 4.0))
       (if (<= x_m 3.4e+56) t_1 (* x_m x_m))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double t_1 = (z * z) * (y * -4.0);
	double tmp;
	if (x_m <= 2.1e-242) {
		tmp = t_1;
	} else if (x_m <= 8e-160) {
		tmp = y * (t * 4.0);
	} else if (x_m <= 3.4e+56) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (z * z) * (y * (-4.0d0))
    if (x_m <= 2.1d-242) then
        tmp = t_1
    else if (x_m <= 8d-160) then
        tmp = y * (t * 4.0d0)
    else if (x_m <= 3.4d+56) then
        tmp = t_1
    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 t_1 = (z * z) * (y * -4.0);
	double tmp;
	if (x_m <= 2.1e-242) {
		tmp = t_1;
	} else if (x_m <= 8e-160) {
		tmp = y * (t * 4.0);
	} else if (x_m <= 3.4e+56) {
		tmp = t_1;
	} else {
		tmp = x_m * x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	t_1 = (z * z) * (y * -4.0)
	tmp = 0
	if x_m <= 2.1e-242:
		tmp = t_1
	elif x_m <= 8e-160:
		tmp = y * (t * 4.0)
	elif x_m <= 3.4e+56:
		tmp = t_1
	else:
		tmp = x_m * x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	t_1 = Float64(Float64(z * z) * Float64(y * -4.0))
	tmp = 0.0
	if (x_m <= 2.1e-242)
		tmp = t_1;
	elseif (x_m <= 8e-160)
		tmp = Float64(y * Float64(t * 4.0));
	elseif (x_m <= 3.4e+56)
		tmp = t_1;
	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)
	t_1 = (z * z) * (y * -4.0);
	tmp = 0.0;
	if (x_m <= 2.1e-242)
		tmp = t_1;
	elseif (x_m <= 8e-160)
		tmp = y * (t * 4.0);
	elseif (x_m <= 3.4e+56)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.1e-242], t$95$1, If[LessEqual[x$95$m, 8e-160], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 3.4e+56], t$95$1, N[(x$95$m * x$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.10000000000000019e-242 or 7.9999999999999999e-160 < x < 3.40000000000000001e56

   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 90.7%

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

   4. Taylor expanded in x around 0 65.0%

      \[\leadsto \color{blue}{-1 \cdot \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(y \cdot {z}^{2}\right)\right)} \]

   5. Taylor expanded in t around 0 44.4%

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

      1. associate-*r* 44.4%

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

      2. *-commutative 44.4%

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

      3. *-commutative 44.4%

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

   7. Simplified 44.4%

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

      1. unpow2 44.4%

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

   9. Applied egg-rr 44.4%

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

   if 2.10000000000000019e-242 < x < 7.9999999999999999e-160

   1. Initial program 93.5%

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

      1. fma-neg 93.5%

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

      2. distribute-lft-neg-in 93.5%

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

      3. *-commutative 93.5%

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

      4. distribute-rgt-neg-in 93.5%

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

      5. metadata-eval 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in t around inf 73.8%

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

      1. *-commutative 73.8%

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

   7. Simplified 73.8%

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

   8. Taylor expanded in y around 0 73.8%

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

      1. associate-*r* 73.9%

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

   10. Simplified 73.9%

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

   if 3.40000000000000001e56 < x

   1. Initial program 77.9%

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

   3. Taylor expanded in y around 0 77.9%

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

   5. Simplified 76.2%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 76.2%

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

   7. Applied egg-rr 76.2%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-242}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+56}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 1.35e+154)
   (+ (* 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 <= 1.35e+154) {
		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 <= 1.35d+154) 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 <= 1.35e+154) {
		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 <= 1.35e+154:
		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 (x_m <= 1.35e+154)
		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 <= 1.35e+154)
		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[x$95$m, 1.35e+154], 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 x < 1.35000000000000003e154

   1. Initial program 93.6%

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

   if 1.35000000000000003e154 < x

   1. Initial program 67.5%

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

   3. Taylor expanded in y around 0 67.5%

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

   5. Simplified 82.5%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 82.5%

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

   7. Applied egg-rr 82.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+176}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* z z) 2e+176)
   (- (* 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 * z) <= 2e+176) {
		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 * z) <= 2d+176) 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 * z) <= 2e+176) {
		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 * z) <= 2e+176:
		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 (Float64(z * z) <= 2e+176)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(z * 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 * z) <= 2e+176)
		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[N[(z * z), $MachinePrecision], 2e+176], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e176

   1. Initial program 96.8%

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

   3. Taylor expanded in z around 0 88.7%

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

      1. *-commutative 88.7%

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

      2. *-commutative 88.7%

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

      3. associate-*l* 88.7%

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

   5. Simplified 88.7%

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

   if 2e176 < (*.f64 z z)

   1. Initial program 77.8%

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

   3. Taylor expanded in z around 0 73.7%

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

   4. Taylor expanded in x around 0 70.0%

      \[\leadsto \color{blue}{-1 \cdot \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(y \cdot {z}^{2}\right)\right)} \]

   5. Taylor expanded in t around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. *-commutative 75.1%

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

   7. Simplified 75.1%

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

      1. unpow2 75.1%

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

   9. Applied egg-rr 75.1%

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

Alternative 5: 58.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 10^{-33}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* x_m x_m) 1e-33) (* 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) <= 1e-33) {
		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) <= 1d-33) 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) <= 1e-33) {
		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) <= 1e-33:
		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) <= 1e-33)
		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) <= 1e-33)
		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], 1e-33], 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) < 1.0000000000000001e-33

   1. Initial program 96.4%

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

   5. Taylor expanded in t around inf 48.2%

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

      1. *-commutative 48.2%

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

   7. Simplified 48.2%

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

   if 1.0000000000000001e-33 < (*.f64 x x)

   1. Initial program 84.4%

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

   3. Taylor expanded in y around 0 84.4%

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

   5. Simplified 71.0%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 71.0%

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

   7. Applied egg-rr 71.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-33}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 2.3e-16) (* 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 <= 2.3e-16) {
		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 <= 2.3d-16) 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 <= 2.3e-16) {
		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 <= 2.3e-16:
		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 (x_m <= 2.3e-16)
		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 <= 2.3e-16)
		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[x$95$m, 2.3e-16], 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 x < 2.2999999999999999e-16

   1. Initial program 92.2%

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

      1. fma-neg 93.4%

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

      2. distribute-lft-neg-in 93.4%

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

      3. *-commutative 93.4%

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

      4. distribute-rgt-neg-in 93.4%

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

      5. metadata-eval 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in t around inf 33.7%

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

      1. *-commutative 33.7%

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

   7. Simplified 33.7%

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

   8. Taylor expanded in y around 0 33.7%

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

      1. associate-*r* 33.7%

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

   10. Simplified 33.7%

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

   if 2.2999999999999999e-16 < x

   1. Initial program 83.3%

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

   3. Taylor expanded in y around 0 83.3%

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

   5. Simplified 62.1%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 62.1%

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

   7. Applied egg-rr 62.1%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.9% 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 89.5%

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

3. Taylor expanded in y around 0 89.5%

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

5. Simplified 47.0%

   \[\leadsto x \cdot x - \color{blue}{0} \]
6. Step-by-step derivation

   1. --rgt-identity 47.0%

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

7. Applied egg-rr 47.0%

   \[\leadsto \color{blue}{x \cdot x} \]
8. Add Preprocessing

Developer Target 1: 90.1% 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 2024130 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

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