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

Percentage Accurate: 90.8% → 93.1%

Time: 9.1s

Alternatives: 6

Speedup: 0.4×

• 48.9% 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: 90.8% 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: 93.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.6 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(x_m, x_m, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot x_m - y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 2.6e+194)
   (fma x_m x_m (* (- (* z z) t) (* y -4.0)))
   (- (* 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 tmp;
	if (x_m <= 2.6e+194) {
		tmp = fma(x_m, x_m, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2.6e+194)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x_m * x_m) - Float64(y * Float64(t * -4.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[x$95$m, 2.6e+194], N[(x$95$m * x$95$m + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 x < 2.5999999999999999e194

   1. Initial program 91.8%

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

      1. fma-neg 93.0%

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

      2. distribute-lft-neg-in 93.0%

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

      3. *-commutative 93.0%

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

      4. distribute-rgt-neg-in 93.0%

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

      5. metadata-eval 93.0%

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

   3. Simplified 93.0%

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

   if 2.5999999999999999e194 < x

   1. Initial program 90.0%

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

   3. Taylor expanded in z around 0 95.0%

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

      1. *-commutative 95.0%

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

      2. *-commutative 95.0%

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

      3. associate-*l* 95.0%

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

   5. Simplified 95.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;x_m \cdot x_m - \left(z \cdot z - t\right) \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{x_m}^{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 3.2e+152)
   (- (* 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 <= 3.2e+152) {
		tmp = (x_m * x_m) - (((z * z) - t) * (y * 4.0));
	} else {
		tmp = pow(x_m, 2.0);
	}
	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 (x_m <= 3.2d+152) then
        tmp = (x_m * x_m) - (((z * z) - t) * (y * 4.0d0))
    else
        tmp = x_m ** 2.0d0
    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 (x_m <= 3.2e+152) {
		tmp = (x_m * x_m) - (((z * z) - t) * (y * 4.0));
	} else {
		tmp = Math.pow(x_m, 2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if x_m <= 3.2e+152:
		tmp = (x_m * x_m) - (((z * z) - t) * (y * 4.0))
	else:
		tmp = math.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 <= 3.2e+152)
		tmp = Float64(Float64(x_m * x_m) - Float64(Float64(Float64(z * z) - t) * Float64(y * 4.0)));
	else
		tmp = x_m ^ 2.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 3.2e+152)
		tmp = (x_m * x_m) - (((z * z) - t) * (y * 4.0));
	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_] := If[LessEqual[x$95$m, 3.2e+152], N[(N[(x$95$m * x$95$m), $MachinePrecision] - 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 < 3.20000000000000005e152

   1. Initial program 92.3%

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

   if 3.20000000000000005e152 < x

   1. Initial program 86.2%

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

   3. Taylor expanded in x around inf 93.1%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;x \cdot x - \left(z \cdot z - t\right) \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	t_1 = (x_m * x_m) - (((z * z) - t) * (y * 4.0));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = (x_m * x_m) - (4.0 * (t * y));
	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[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.7%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.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. Step-by-step derivation

      1. sub-neg 0.0%

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

      2. flip-+ 0.0%

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

      3. pow2 0.0%

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

      4. pow2 0.0%

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

      5. pow-prod-up 0.0%

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

      6. metadata-eval 0.0%

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

      7. pow2 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. associate-*r/ 0.0%

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

      2. clear-num 0.0%

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

      3. unpow2 0.0%

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

      4. fma-neg 0.0%

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

      5. remove-double-neg 0.0%

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

      6. *-commutative 0.0%

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

      7. associate-*l* 0.0%

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

      8. metadata-eval 0.0%

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

      9. pow-prod-up 0.0%

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

      10. sqr-neg 0.0%

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

      11. difference-of-squares 0.0%

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

      12. remove-double-neg 0.0%

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

      13. sub-neg 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-prod 0.0%

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

      16. sqr-neg 0.0%

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

      17. sqrt-prod 0.0%

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

      18. add-sqr-sqrt 9.1%

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

   6. Applied egg-rr 9.1%

      \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z, z, t\right)}{4 \cdot \left(y \cdot {\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}\right)}}} \]
   7. Step-by-step derivation

      1. *-lft-identity 9.1%

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

      2. times-frac 9.1%

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

      3. metadata-eval 9.1%

         \[\leadsto x \cdot x - \frac{1}{\color{blue}{0.25} \cdot \frac{\mathsf{fma}\left(z, z, t\right)}{y \cdot {\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}}} \]

      4. *-commutative 9.1%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\mathsf{fma}\left(z, z, t\right)}{\color{blue}{{\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2} \cdot y}}} \]

      5. associate-/r* 9.1%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(z, z, t\right)}{{\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}}}{y}}} \]

      6. remove-double-div 9.1%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{\color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, z, t\right)}}}}{{\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}}}{y}} \]

      7. associate-/r* 9.1%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, z, t\right)} \cdot {\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}}}}{y}} \]

      8. *-commutative 9.1%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{1}{\color{blue}{{\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2} \cdot \frac{1}{\mathsf{fma}\left(z, z, t\right)}}}}{y}} \]

      9. unpow2 9.1%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{1}{\color{blue}{\left(\mathsf{fma}\left(z, z, t\right) \cdot \mathsf{fma}\left(z, z, t\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(z, z, t\right)}}}{y}} \]

      10. associate-*r* 9.1%

          \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(z, z, t\right) \cdot \left(\mathsf{fma}\left(z, z, t\right) \cdot \frac{1}{\mathsf{fma}\left(z, z, t\right)}\right)}}}{y}} \]

      11. rgt-mult-inverse 9.1%

          \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{1}{\mathsf{fma}\left(z, z, t\right) \cdot \color{blue}{1}}}{y}} \]

      12. *-rgt-identity 9.1%

          \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(z, z, t\right)}}}{y}} \]

   8. Simplified 9.1%

      \[\leadsto x \cdot x - \color{blue}{\frac{1}{0.25 \cdot \frac{\frac{1}{\mathsf{fma}\left(z, z, t\right)}}{y}}} \]

   9. Taylor expanded in z around 0 63.6%

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

       1. *-commutative 63.6%

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

   11. Simplified 63.6%

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

4. Final simplification 94.4%

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

Alternative 4: 57.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_1 := 4 \cdot \left(t \cdot y\right)\\ \mathbf{if}\;x_m \cdot x_m \leq 1.8 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot x_m - t_1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (let* ((t_1 (* 4.0 (* t y))))
   (if (<= (* x_m x_m) 1.8e+105) t_1 (- (* x_m x_m) t_1))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double t_1 = 4.0 * (t * y);
	double tmp;
	if ((x_m * x_m) <= 1.8e+105) {
		tmp = t_1;
	} else {
		tmp = (x_m * x_m) - t_1;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z, t)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 4.0d0 * (t * y)
    if ((x_m * x_m) <= 1.8d+105) then
        tmp = t_1
    else
        tmp = (x_m * x_m) - t_1
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double t_1 = 4.0 * (t * y);
	double tmp;
	if ((x_m * x_m) <= 1.8e+105) {
		tmp = t_1;
	} else {
		tmp = (x_m * x_m) - t_1;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	t_1 = 4.0 * (t * y)
	tmp = 0
	if (x_m * x_m) <= 1.8e+105:
		tmp = t_1
	else:
		tmp = (x_m * x_m) - t_1
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	t_1 = Float64(4.0 * Float64(t * y))
	tmp = 0.0
	if (Float64(x_m * x_m) <= 1.8e+105)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m * x_m) - t_1);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	t_1 = 4.0 * (t * y);
	tmp = 0.0;
	if ((x_m * x_m) <= 1.8e+105)
		tmp = t_1;
	else
		tmp = (x_m * x_m) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 1.8e+105], t$95$1, N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.7999999999999999e105

   1. Initial program 94.6%

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

   3. Taylor expanded in t around inf 50.2%

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

      1. *-commutative 50.2%

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

   5. Simplified 50.2%

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

   if 1.7999999999999999e105 < (*.f64 x x)

   1. Initial program 86.8%

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

      1. sub-neg 86.8%

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

      2. flip-+ 53.4%

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

      3. pow2 53.4%

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

      4. pow2 53.4%

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

      5. pow-prod-up 53.4%

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

      6. metadata-eval 53.4%

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

      7. pow2 53.4%

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

   4. Applied egg-rr 53.4%

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

      1. associate-*r/ 50.6%

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

      2. clear-num 50.6%

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

      3. unpow2 50.6%

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

      4. fma-neg 50.6%

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

      5. remove-double-neg 50.6%

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

      6. *-commutative 50.6%

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

      7. associate-*l* 50.6%

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

      8. metadata-eval 50.6%

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

      9. pow-prod-up 50.6%

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

      10. sqr-neg 50.6%

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

      11. difference-of-squares 51.7%

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

      12. remove-double-neg 51.7%

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

      13. sub-neg 51.7%

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

      14. add-sqr-sqrt 25.9%

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

      15. sqrt-prod 49.5%

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

      16. sqr-neg 49.5%

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

      17. sqrt-prod 24.6%

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

      18. add-sqr-sqrt 47.1%

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

   6. Applied egg-rr 47.1%

      \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z, z, t\right)}{4 \cdot \left(y \cdot {\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}\right)}}} \]
   7. Step-by-step derivation

      1. *-lft-identity 47.1%

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

      2. times-frac 47.1%

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

      3. metadata-eval 47.1%

         \[\leadsto x \cdot x - \frac{1}{\color{blue}{0.25} \cdot \frac{\mathsf{fma}\left(z, z, t\right)}{y \cdot {\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}}} \]

      4. *-commutative 47.1%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\mathsf{fma}\left(z, z, t\right)}{\color{blue}{{\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2} \cdot y}}} \]

      5. associate-/r* 48.2%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(z, z, t\right)}{{\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}}}{y}}} \]

      6. remove-double-div 48.2%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{\color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, z, t\right)}}}}{{\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}}}{y}} \]

      7. associate-/r* 48.2%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, z, t\right)} \cdot {\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}}}}{y}} \]

      8. *-commutative 48.2%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{1}{\color{blue}{{\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2} \cdot \frac{1}{\mathsf{fma}\left(z, z, t\right)}}}}{y}} \]

      9. unpow2 48.2%

         \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{1}{\color{blue}{\left(\mathsf{fma}\left(z, z, t\right) \cdot \mathsf{fma}\left(z, z, t\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(z, z, t\right)}}}{y}} \]

      10. associate-*r* 61.2%

          \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(z, z, t\right) \cdot \left(\mathsf{fma}\left(z, z, t\right) \cdot \frac{1}{\mathsf{fma}\left(z, z, t\right)}\right)}}}{y}} \]

      11. rgt-mult-inverse 78.7%

          \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{1}{\mathsf{fma}\left(z, z, t\right) \cdot \color{blue}{1}}}{y}} \]

      12. *-rgt-identity 78.7%

          \[\leadsto x \cdot x - \frac{1}{0.25 \cdot \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(z, z, t\right)}}}{y}} \]

   8. Simplified 78.7%

      \[\leadsto x \cdot x - \color{blue}{\frac{1}{0.25 \cdot \frac{\frac{1}{\mathsf{fma}\left(z, z, t\right)}}{y}}} \]

   9. Taylor expanded in z around 0 75.0%

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

       1. *-commutative 75.0%

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

   11. Simplified 75.0%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.8 \cdot 10^{+105}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - 4 \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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) - (y * (t * (-4.0d0)))
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) - (y * (t * -4.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $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 z around 0 68.5%

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

   1. *-commutative 68.5%

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

   2. *-commutative 68.5%

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

   3. associate-*l* 68.5%

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

5. Simplified 68.5%

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

6. Final simplification 68.5%

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

Alternative 6: 31.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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 * (t * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := N[(4.0 * N[(t * y), $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 36.1%

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

   1. *-commutative 36.1%

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

5. Simplified 36.1%

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

6. Final simplification 36.1%

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

Developer target: 90.8% 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 2024021 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

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