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

Percentage Accurate: 90.7% → 93.7%

Time: 10.4s

Alternatives: 8

Speedup: 0.7×

• 48.4% 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: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{+249}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 1e+249) (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 <= 1e+249) {
		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 <= 1e+249)
		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, 1e+249], 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 < 9.9999999999999992e248

   1. Initial program 89.9%

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

      1. fmm-def 92.8%

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

      2. distribute-lft-neg-in 92.8%

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

      3. *-commutative 92.8%

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

      4. distribute-rgt-neg-in 92.8%

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

      5. metadata-eval 92.8%

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

   3. Simplified 92.8%

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

   if 9.9999999999999992e248 < x

   1. Initial program 68.8%

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

   3. Taylor expanded in y around 0 68.8%

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

   5. Simplified 100.0%

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

      1. --rgt-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

Alternative 2: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+31}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + \frac{z \cdot -4}{\frac{t}{z}}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* z z) 1e+31)
   (- (* x_m x_m) (* y (* t -4.0)))
   (* t (* y (+ 4.0 (/ (* z -4.0) (/ t z)))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+31) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z, t)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 1d+31) then
        tmp = (x_m * x_m) - (y * (t * (-4.0d0)))
    else
        tmp = t * (y * (4.0d0 + ((z * (-4.0d0)) / (t / z))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+31) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if (z * z) <= 1e+31:
		tmp = (x_m * x_m) - (y * (t * -4.0))
	else:
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+31)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(t * Float64(y * Float64(4.0 + Float64(Float64(z * -4.0) / Float64(t / z)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+31)
		tmp = (x_m * x_m) - (y * (t * -4.0));
	else
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))));
	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], 1e+31], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999996e30

   1. Initial program 95.4%

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

   3. Taylor expanded in z around 0 90.7%

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

      1. *-commutative 90.7%

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

      2. *-commutative 90.7%

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

      3. associate-*l* 90.7%

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

   5. Simplified 90.7%

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

   if 9.9999999999999996e30 < (*.f64 z z)

   1. Initial program 81.3%

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

   3. Taylor expanded in t around inf 75.1%

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

   4. Taylor expanded in y around -inf 71.3%

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

      1. unpow2 71.3%

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

      2. *-un-lft-identity 71.3%

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

      3. times-frac 72.8%

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

   6. Applied egg-rr 72.8%

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

      1. /-rgt-identity 72.8%

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

      2. associate-*r* 72.8%

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

      3. clear-num 72.9%

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

      4. un-div-inv 72.9%

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

   8. Applied egg-rr 72.9%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+31}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + \frac{z \cdot -4}{\frac{t}{z}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+31}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + \left(z \cdot -4\right) \cdot \frac{z}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* z z) 1e+31)
   (- (* x_m x_m) (* y (* t -4.0)))
   (* t (* y (+ 4.0 (* (* z -4.0) (/ z t)))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+31) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = t * (y * (4.0 + ((z * -4.0) * (z / t))));
	}
	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) <= 1d+31) then
        tmp = (x_m * x_m) - (y * (t * (-4.0d0)))
    else
        tmp = t * (y * (4.0d0 + ((z * (-4.0d0)) * (z / t))))
    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) <= 1e+31) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = t * (y * (4.0 + ((z * -4.0) * (z / t))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if (z * z) <= 1e+31:
		tmp = (x_m * x_m) - (y * (t * -4.0))
	else:
		tmp = t * (y * (4.0 + ((z * -4.0) * (z / t))))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+31)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(t * Float64(y * Float64(4.0 + Float64(Float64(z * -4.0) * Float64(z / t)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+31)
		tmp = (x_m * x_m) - (y * (t * -4.0));
	else
		tmp = t * (y * (4.0 + ((z * -4.0) * (z / t))));
	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], 1e+31], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999996e30

   1. Initial program 95.4%

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

   3. Taylor expanded in z around 0 90.7%

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

      1. *-commutative 90.7%

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

      2. *-commutative 90.7%

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

      3. associate-*l* 90.7%

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

   5. Simplified 90.7%

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

   if 9.9999999999999996e30 < (*.f64 z z)

   1. Initial program 81.3%

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

   3. Taylor expanded in t around inf 75.1%

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

   4. Taylor expanded in y around -inf 71.3%

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

      1. clear-num 71.3%

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

      2. un-div-inv 71.3%

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

   6. Applied egg-rr 71.3%

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

      1. pow2 71.3%

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

      2. div-inv 71.3%

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

      3. clear-num 71.3%

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

      4. associate-*r/ 72.8%

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

      5. associate-*r* 72.8%

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

   8. Applied egg-rr 72.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+31}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + \left(z \cdot -4\right) \cdot \frac{z}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.7 \cdot 10^{+35}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + -4 \cdot \frac{z \cdot z}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* z z) 4.7e+35)
   (- (* x_m x_m) (* y (* t -4.0)))
   (* t (* y (+ 4.0 (* -4.0 (/ (* z z) t)))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4.7e+35) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = t * (y * (4.0 + (-4.0 * ((z * z) / t))));
	}
	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) <= 4.7d+35) then
        tmp = (x_m * x_m) - (y * (t * (-4.0d0)))
    else
        tmp = t * (y * (4.0d0 + ((-4.0d0) * ((z * z) / t))))
    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) <= 4.7e+35) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = t * (y * (4.0 + (-4.0 * ((z * z) / t))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if (z * z) <= 4.7e+35:
		tmp = (x_m * x_m) - (y * (t * -4.0))
	else:
		tmp = t * (y * (4.0 + (-4.0 * ((z * z) / t))))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4.7e+35)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(t * Float64(y * Float64(4.0 + Float64(-4.0 * Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 4.7e+35)
		tmp = (x_m * x_m) - (y * (t * -4.0));
	else
		tmp = t * (y * (4.0 + (-4.0 * ((z * z) / t))));
	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], 4.7e+35], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(4.0 + N[(-4.0 * N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.70000000000000033e35

   1. Initial program 95.4%

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

   3. Taylor expanded in z around 0 90.7%

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

      1. *-commutative 90.7%

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

      2. *-commutative 90.7%

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

      3. associate-*l* 90.7%

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

   5. Simplified 90.7%

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

   if 4.70000000000000033e35 < (*.f64 z z)

   1. Initial program 81.3%

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

   3. Taylor expanded in t around inf 75.1%

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

   4. Taylor expanded in y around -inf 71.3%

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

      1. unpow2 71.3%

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

   6. Applied egg-rr 71.3%

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

Alternative 5: 93.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.8 \cdot 10^{+149}:\\ \;\;\;\;x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 4.8e+149)
   (+ (* 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 <= 4.8e+149) {
		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 <= 4.8d+149) 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 <= 4.8e+149) {
		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 <= 4.8e+149:
		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 <= 4.8e+149)
		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 <= 4.8e+149)
		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, 4.8e+149], 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 < 4.80000000000000024e149

   1. Initial program 90.3%

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

   if 4.80000000000000024e149 < x

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0 78.4%

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

   5. Simplified 94.6%

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

      1. --rgt-identity 94.6%

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

   7. Applied egg-rr 94.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+149}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{+149}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 4e+149) (- (* x_m x_m) (* 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 <= 4e+149) {
		tmp = (x_m * x_m) - (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 <= 4d+149) then
        tmp = (x_m * x_m) - (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 <= 4e+149) {
		tmp = (x_m * x_m) - (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 <= 4e+149:
		tmp = (x_m * x_m) - (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 <= 4e+149)
		tmp = Float64(Float64(x_m * x_m) - 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 <= 4e+149)
		tmp = (x_m * x_m) - (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, 4e+149], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0000000000000002e149

   1. Initial program 90.3%

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

   3. Taylor expanded in z around 0 59.5%

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

      1. *-commutative 59.5%

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

      2. *-commutative 59.5%

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

      3. associate-*l* 59.5%

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

   5. Simplified 59.5%

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

   if 4.0000000000000002e149 < x

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0 78.4%

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

   5. Simplified 94.6%

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

      1. --rgt-identity 94.6%

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

   7. Applied egg-rr 94.6%

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

Alternative 7: 58.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.9 \cdot 10^{+47}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 4.9e+47) (* 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 <= 4.9e+47) {
		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 <= 4.9d+47) 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 <= 4.9e+47) {
		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 <= 4.9e+47:
		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 (x_m <= 4.9e+47)
		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 <= 4.9e+47)
		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[x$95$m, 4.9e+47], 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 x < 4.9000000000000003e47

   1. Initial program 90.1%

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

      1. fmm-def 92.6%

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

      2. distribute-lft-neg-in 92.6%

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

      3. *-commutative 92.6%

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

      4. distribute-rgt-neg-in 92.6%

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

      5. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in t around inf 36.8%

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

      1. *-commutative 36.8%

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

   7. Simplified 36.8%

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

   if 4.9000000000000003e47 < x

   1. Initial program 82.4%

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

   3. Taylor expanded in y around 0 82.4%

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

   5. Simplified 83.7%

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

      1. --rgt-identity 83.7%

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

   7. Applied egg-rr 83.7%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{+47}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.7% 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 88.6%

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

3. Taylor expanded in y around 0 88.6%

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

5. Simplified 39.7%

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

   1. --rgt-identity 39.7%

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

7. Applied egg-rr 39.7%

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

Developer Target 1: 90.7% 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 2024180 
(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))))