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

Percentage Accurate: 90.7% → 93.9%

Time: 11.4s

Alternatives: 8

Speedup: 0.4×

• 49.2% 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: 93.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 2e+240) (fma x_m x_m (* (- (* z z) t) (* y -4.0))) (pow x_m 2.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2e+240) {
		tmp = fma(x_m, x_m, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = pow(x_m, 2.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2e+240)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = x_m ^ 2.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[x$95$m, 2e+240], N[(x$95$m * x$95$m + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$95$m, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000003e240

   1. Initial program 92.9%

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

      1. fma-neg 94.2%

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

      2. distribute-lft-neg-in 94.2%

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

      3. *-commutative 94.2%

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

      4. distribute-rgt-neg-in 94.2%

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

      5. metadata-eval 94.2%

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

   3. Simplified 94.2%

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

   if 2.00000000000000003e240 < x

   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 x around inf 100.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(y \cdot 4, \mathsf{fma}\left(z, -z, t\right), x\_m \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (fma (* y 4.0) (fma z (- z) t) (* x_m x_m)))

C

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

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	return fma(Float64(y * 4.0), fma(z, Float64(-z), t), Float64(x_m * x_m))
end

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. cancel-sign-sub-inv 91.6%

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

   2. distribute-lft-neg-out 91.6%

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

   3. +-commutative 91.6%

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

   4. distribute-lft-neg-out 91.6%

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

   5. distribute-lft-neg-in 91.6%

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

   6. distribute-rgt-neg-in 91.6%

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

   7. fma-define 93.6%

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

   8. neg-sub0 93.6%

      \[\leadsto \mathsf{fma}\left(y \cdot 4, \color{blue}{0 - \left(z \cdot z - t\right)}, x \cdot x\right) \]

   9. associate-+l- 93.6%

      \[\leadsto \mathsf{fma}\left(y \cdot 4, \color{blue}{\left(0 - z \cdot z\right) + t}, x \cdot x\right) \]

   10. neg-sub0 93.6%

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

   11. distribute-rgt-neg-out 93.6%

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

   12. fma-define 93.6%

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

3. Simplified 93.6%

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

5. Final simplification 93.6%

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

Alternative 3: 93.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_1 := x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (let* ((t_1 (+ (* x_m x_m) (* (* y 4.0) (- t (* z z))))))
   (if (<= t_1 INFINITY) t_1 (pow x_m 2.0))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double t_1 = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = pow(x_m, 2.0);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double t_1 = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = Math.pow(x_m, 2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	t_1 = (x_m * x_m) + ((y * 4.0) * (t - (z * z)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = math.pow(x_m, 2.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	t_1 = Float64(Float64(x_m * x_m) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = x_m ^ 2.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	t_1 = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = x_m ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[Power[x$95$m, 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 95.3%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf 70.0%

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

4. Final simplification 94.4%

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

Alternative 4: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(y \cdot 4, t - z \cdot z, x\_m \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (fma (* 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) {
	return fma((y * 4.0), (t - (z * z)), (x_m * x_m));
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	return fma(Float64(y * 4.0), Float64(t - Float64(z * z)), Float64(x_m * x_m))
end

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. cancel-sign-sub-inv 91.6%

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

   2. distribute-lft-neg-out 91.6%

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

   3. +-commutative 91.6%

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

   4. distribute-lft-neg-out 91.6%

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

   5. distribute-lft-neg-in 91.6%

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

   6. distribute-rgt-neg-in 91.6%

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

   7. fma-define 93.6%

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

   8. sub-neg 93.6%

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

   9. +-commutative 93.6%

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

   10. distribute-neg-in 93.6%

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

   11. remove-double-neg 93.6%

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

   12. sub-neg 93.6%

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

3. Simplified 93.6%

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

5. Final simplification 93.6%

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

Alternative 5: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+22} \lor \neg \left(z \cdot z \leq 5 \cdot 10^{+38}\right) \land z \cdot z \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (or (<= (* z z) 2e+22) (and (not (<= (* z z) 5e+38)) (<= (* z z) 2e+225)))
   (- (* 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+22) || (!((z * z) <= 5e+38) && ((z * z) <= 2e+225))) {
		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+22) .or. (.not. ((z * z) <= 5d+38)) .and. ((z * z) <= 2d+225)) 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+22) || (!((z * z) <= 5e+38) && ((z * z) <= 2e+225))) {
		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+22) or (not ((z * z) <= 5e+38) and ((z * z) <= 2e+225)):
		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+22) || (!(Float64(z * z) <= 5e+38) && (Float64(z * z) <= 2e+225)))
		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+22) || (~(((z * z) <= 5e+38)) && ((z * z) <= 2e+225)))
		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[Or[LessEqual[N[(z * z), $MachinePrecision], 2e+22], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 5e+38]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 2e+225]]], 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) < 2e22 or 4.9999999999999997e38 < (*.f64 z z) < 1.99999999999999986e225

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0 90.2%

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

      1. *-commutative 90.2%

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

      2. *-commutative 90.2%

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

      3. associate-*l* 90.2%

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

   5. Simplified 90.2%

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

   if 2e22 < (*.f64 z z) < 4.9999999999999997e38 or 1.99999999999999986e225 < (*.f64 z z)

   1. Initial program 78.2%

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

   3. Taylor expanded in z around inf 76.3%

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

      1. associate-*r* 76.3%

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

      2. *-commutative 76.3%

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

      3. *-commutative 76.3%

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

   5. Simplified 76.3%

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

      1. unpow2 76.3%

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

   7. Applied egg-rr 76.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+22} \lor \neg \left(z \cdot z \leq 5 \cdot 10^{+38}\right) \land z \cdot z \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_1 := x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (let* ((t_1 (+ (* x_m x_m) (* (* y 4.0) (- t (* z z))))))
   (if (<= t_1 INFINITY) t_1 (- (* x_m x_m) (* y (* t -4.0))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double t_1 = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double t_1 = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	t_1 = (x_m * x_m) + ((y * 4.0) * (t - (z * z)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = (x_m * x_m) - (y * (t * -4.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	t_1 = Float64(Float64(x_m * x_m) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m * x_m) - Float64(y * Float64(t * -4.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	t_1 = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = (x_m * x_m) - (y * (t * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 95.3%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 50.0%

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

      1. *-commutative 50.0%

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

      2. *-commutative 50.0%

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

      3. associate-*l* 50.0%

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

   5. Simplified 50.0%

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

4. Final simplification 93.6%

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

Alternative 7: 56.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2050000000:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* z z) 2050000000.0) (* 4.0 (* y t)) (* (* 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) <= 2050000000.0) {
		tmp = 4.0 * (y * t);
	} 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) <= 2050000000.0d0) then
        tmp = 4.0d0 * (y * t)
    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) <= 2050000000.0) {
		tmp = 4.0 * (y * t);
	} 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) <= 2050000000.0:
		tmp = 4.0 * (y * t)
	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) <= 2050000000.0)
		tmp = Float64(4.0 * Float64(y * t));
	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) <= 2050000000.0)
		tmp = 4.0 * (y * t);
	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], 2050000000.0], N[(4.0 * N[(y * t), $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) < 2.05e9

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 43.4%

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

      1. *-commutative 43.4%

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

   5. Simplified 43.4%

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

   if 2.05e9 < (*.f64 z z)

   1. Initial program 82.8%

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

   3. Taylor expanded in z around inf 61.2%

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

      1. associate-*r* 61.2%

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

      2. *-commutative 61.2%

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

      3. *-commutative 61.2%

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

   5. Simplified 61.2%

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

      1. unpow2 61.2%

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

   7. Applied egg-rr 61.2%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2050000000:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.2% accurate, 2.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t) :precision binary64 (* 4.0 (* y t)))

C

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

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 = 4.0d0 * (y * t)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	return 4.0 * (y * t);
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	return 4.0 * (y * t)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in t around inf 28.7%

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

   1. *-commutative 28.7%

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

5. Simplified 28.7%

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

6. Final simplification 28.7%

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

Developer target: 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 2024077 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

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

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