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

Percentage Accurate: 90.5% → 93.8%

Time: 11.4s

Alternatives: 7

Speedup: 0.6×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1 \cdot 10^{+240}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 1.1e+240) (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 <= 1.1e+240) {
		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 <= 1.1e+240)
		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, 1.1e+240], 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 < 1.1000000000000001e240

   1. Initial program 93.2%

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

      1. fma-neg 95.3%

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

      2. distribute-lft-neg-in 95.3%

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

      3. *-commutative 95.3%

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

      4. distribute-rgt-neg-in 95.3%

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

      5. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

   if 1.1000000000000001e240 < x

   1. Initial program 66.7%

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

   3. Taylor expanded in y around 0 66.7%

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

   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: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+262}:\\ \;\;\;\;x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m - t \cdot \left(4 \cdot \left(z \cdot \frac{z \cdot y}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* z z) 2e+262)
   (+ (* x_m x_m) (* (* y 4.0) (- t (* z z))))
   (- (* x_m x_m) (* t (* 4.0 (* z (/ (* z y) t)))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+262) {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x_m * x_m) - (t * (4.0 * (z * ((z * y) / 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) <= 2d+262) then
        tmp = (x_m * x_m) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x_m * x_m) - (t * (4.0d0 * (z * ((z * y) / 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) <= 2e+262) {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x_m * x_m) - (t * (4.0 * (z * ((z * y) / t))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if (z * z) <= 2e+262:
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x_m * x_m) - (t * (4.0 * (z * ((z * y) / t))))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+262)
		tmp = Float64(Float64(x_m * x_m) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x_m * x_m) - Float64(t * Float64(4.0 * Float64(z * Float64(Float64(z * y) / 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) <= 2e+262)
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x_m * x_m) - (t * (4.0 * (z * ((z * y) / 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], 2e+262], 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[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(t * N[(4.0 * N[(z * N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e262

   1. Initial program 98.2%

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

   if 2e262 < (*.f64 z z)

   1. Initial program 78.8%

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

   3. Taylor expanded in t around inf 78.8%

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

      1. +-commutative 78.8%

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

      2. *-commutative 78.8%

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

      3. *-commutative 78.8%

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

      4. metadata-eval 78.8%

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

      5. distribute-rgt-neg-in 78.8%

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

      6. distribute-lft-neg-in 78.8%

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

      7. distribute-rgt-out 78.8%

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

      8. unsub-neg 78.8%

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

      9. associate-/l* 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in z around inf 78.8%

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

      1. associate-*l/ 74.9%

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

      2. *-commutative 74.9%

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

   8. Simplified 74.9%

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

      1. associate-*r/ 78.8%

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

      2. *-commutative 78.8%

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

      3. div-inv 78.8%

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

      4. unpow2 78.8%

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

      5. associate-*r* 88.4%

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

      6. associate-*r* 86.0%

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

      7. div-inv 86.0%

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

      8. clear-num 86.0%

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

      9. un-div-inv 86.7%

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

   10. Applied egg-rr 86.7%

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

       1. associate-/r/ 88.4%

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

   12. Applied egg-rr 88.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+262}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - t \cdot \left(4 \cdot \left(z \cdot \frac{z \cdot y}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 5.6 \cdot 10^{+307}:\\ \;\;\;\;x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* x_m x_m) 5.6e+307)
   (+ (* 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 * x_m) <= 5.6e+307) {
		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 * x_m) <= 5.6d+307) 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 * x_m) <= 5.6e+307) {
		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 * x_m) <= 5.6e+307:
		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 (Float64(x_m * x_m) <= 5.6e+307)
		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 * x_m) <= 5.6e+307)
		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[N[(x$95$m * x$95$m), $MachinePrecision], 5.6e+307], 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 (*.f64 x x) < 5.6000000000000002e307

   1. Initial program 96.1%

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

   if 5.6000000000000002e307 < (*.f64 x x)

   1. Initial program 79.7%

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

   3. Taylor expanded in y around 0 79.7%

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

   5. Simplified 91.5%

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

      1. --rgt-identity 91.5%

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

   7. Applied egg-rr 91.5%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.6 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{-248}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 3.3e-248)
   (* 4.0 (* t y))
   (if (<= z 1.7e+85) (* x_m x_m) (* -4.0 (* (* z z) y)))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 3.3e-248) {
		tmp = 4.0 * (t * y);
	} else if (z <= 1.7e+85) {
		tmp = x_m * x_m;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	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 <= 3.3d-248) then
        tmp = 4.0d0 * (t * y)
    else if (z <= 1.7d+85) then
        tmp = x_m * x_m
    else
        tmp = (-4.0d0) * ((z * z) * y)
    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 <= 3.3e-248) {
		tmp = 4.0 * (t * y);
	} else if (z <= 1.7e+85) {
		tmp = x_m * x_m;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if z <= 3.3e-248:
		tmp = 4.0 * (t * y)
	elif z <= 1.7e+85:
		tmp = x_m * x_m
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 3.3e-248)
		tmp = Float64(4.0 * Float64(t * y));
	elseif (z <= 1.7e+85)
		tmp = Float64(x_m * x_m);
	else
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if (z <= 3.3e-248)
		tmp = 4.0 * (t * y);
	elseif (z <= 1.7e+85)
		tmp = x_m * x_m;
	else
		tmp = -4.0 * ((z * z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 3.3e-248], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.7e+85], N[(x$95$m * x$95$m), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.3000000000000002e-248

   1. Initial program 94.5%

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

      1. fma-neg 97.2%

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

      2. distribute-lft-neg-in 97.2%

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

      3. *-commutative 97.2%

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

      4. distribute-rgt-neg-in 97.2%

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

      5. metadata-eval 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in t around inf 36.0%

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

      1. *-commutative 36.0%

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

   7. Simplified 36.0%

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

   if 3.3000000000000002e-248 < z < 1.7000000000000002e85

   1. Initial program 98.3%

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

   3. Taylor expanded in y around 0 98.3%

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

   5. Simplified 62.4%

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

      1. --rgt-identity 62.4%

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

   7. Applied egg-rr 62.4%

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

   if 1.7000000000000002e85 < z

   1. Initial program 78.3%

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

   3. Taylor expanded in t around inf 76.2%

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

   4. Taylor expanded in z around inf 72.5%

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

      1. *-commutative 72.5%

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

   6. Simplified 72.5%

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

      1. unpow2 72.5%

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

   8. Applied egg-rr 72.5%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{-248}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 2.25 \cdot 10^{+89}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 2.25e+89) (- (* x_m x_m) (* y (* t -4.0))) (* -4.0 (* (* z z) y))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 2.25e+89) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	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 <= 2.25d+89) then
        tmp = (x_m * x_m) - (y * (t * (-4.0d0)))
    else
        tmp = (-4.0d0) * ((z * z) * y)
    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 <= 2.25e+89) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if z <= 2.25e+89:
		tmp = (x_m * x_m) - (y * (t * -4.0))
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 2.25e+89)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if (z <= 2.25e+89)
		tmp = (x_m * x_m) - (y * (t * -4.0));
	else
		tmp = -4.0 * ((z * z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 2.25e+89], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.25e89

   1. Initial program 95.7%

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

   3. Taylor expanded in z around 0 75.4%

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

      1. *-commutative 75.4%

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

      2. *-commutative 75.4%

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

      3. associate-*l* 75.4%

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

   5. Simplified 75.4%

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

   if 2.25e89 < z

   1. Initial program 76.9%

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

   3. Taylor expanded in t around inf 74.7%

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

   4. Taylor expanded in z around inf 74.8%

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

      1. *-commutative 74.8%

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

   6. Simplified 74.8%

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

      1. unpow2 74.8%

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

   8. Applied egg-rr 74.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.25 \cdot 10^{+89}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* x_m x_m) 1.4e+25) (* 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) <= 1.4e+25) {
		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) <= 1.4d+25) 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) <= 1.4e+25) {
		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) <= 1.4e+25:
		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) <= 1.4e+25)
		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) <= 1.4e+25)
		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], 1.4e+25], 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) < 1.4000000000000001e25

   1. Initial program 96.1%

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

      1. fma-neg 96.1%

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

      2. distribute-lft-neg-in 96.1%

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

      3. *-commutative 96.1%

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

      4. distribute-rgt-neg-in 96.1%

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

      5. metadata-eval 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in t around inf 50.1%

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

      1. *-commutative 50.1%

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

   7. Simplified 50.1%

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

   if 1.4000000000000001e25 < (*.f64 x x)

   1. Initial program 87.2%

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

   3. Taylor expanded in y around 0 87.2%

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

   5. Simplified 75.9%

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

      1. --rgt-identity 75.9%

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

   7. Applied egg-rr 75.9%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.6% 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 92.3%

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

3. Taylor expanded in y around 0 92.3%

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

5. Simplified 40.6%

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

   1. --rgt-identity 40.6%

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

7. Applied egg-rr 40.6%

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

Developer Target 1: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024133 
(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))))