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

Percentage Accurate: 90.5% → 93.0%

Time: 11.0s

Alternatives: 10

Speedup: 0.4×

• 48.7% 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: 93.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m - t \cdot \left(y \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 1.45e+246)
   (fma x_m x_m (* (- (* z z) t) (* y -4.0)))
   (- (* x_m x_m) (* t (* y 4.0)))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 1.45e+246) {
		tmp = fma(x_m, x_m, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = (x_m * x_m) - (t * (y * 4.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 1.45e+246)
		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(t * Float64(y * 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, 1.45e+246], 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[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45000000000000007e246

   1. Initial program 88.9%

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

      1. fma-neg 92.6%

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

      2. distribute-lft-neg-in 92.6%

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

      3. *-commutative 92.6%

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

      4. distribute-rgt-neg-in 92.6%

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

      5. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

   if 1.45000000000000007e246 < x

   1. Initial program 75.0%

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

   3. Taylor expanded in t around inf 75.0%

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

      1. associate-*r* 75.0%

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

      2. sub-neg 75.0%

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

      3. distribute-rgt-in 75.0%

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

      4. *-commutative 75.0%

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

      5. *-commutative 75.0%

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

      6. add-sqr-sqrt 50.0%

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

      7. sqrt-unprod 66.7%

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

      8. swap-sqr 66.7%

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

      9. metadata-eval 66.7%

         \[\leadsto x \cdot x - \left(\frac{{z}^{2}}{t} \cdot \left(t \cdot \left(y \cdot 4\right)\right) + \left(-1\right) \cdot \left(t \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}}\right)\right) \]

      10. metadata-eval 66.7%

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

      11. swap-sqr 66.7%

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

      12. sqrt-unprod 33.3%

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

      13. add-sqr-sqrt 83.3%

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

      14. associate-*l* 83.3%

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

      15. metadata-eval 83.3%

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

      16. neg-mul-1 83.3%

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

      17. add-sqr-sqrt 33.3%

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

   5. Applied egg-rr 83.3%

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

      1. distribute-lft1-in 91.7%

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

      2. unpow2 91.7%

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

      3. associate-*r/ 91.7%

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

      4. fma-define 91.7%

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

   7. Simplified 91.7%

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

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto x \cdot x - \color{blue}{1} \cdot \left(t \cdot \left(y \cdot 4\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left(z \cdot z - t\right) \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{+252}:\\ \;\;\;\;x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* (- (* z z) t) (* y 4.0)) 4e+252)
   (+ (* x_m x_m) (* (* y 4.0) (- t (* z z))))
   (* -4.0 (* t (* y (fma z (/ z t) -1.0))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((((z * z) - t) * (y * 4.0)) <= 4e+252) {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = -4.0 * (t * (y * fma(z, (z / t), -1.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(z * z) - t) * Float64(y * 4.0)) <= 4e+252)
		tmp = Float64(Float64(x_m * x_m) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(y * fma(z, Float64(z / t), -1.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 4e+252], 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[(-4.0 * N[(t * N[(y * N[(z * N[(z / t), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)) < 4.0000000000000004e252

   1. Initial program 97.1%

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

   if 4.0000000000000004e252 < (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))

   1. Initial program 58.5%

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

   3. Taylor expanded in t around inf 58.5%

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

   4. Taylor expanded in x around 0 73.7%

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

      1. unpow2 73.7%

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

      2. associate-*r/ 76.9%

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

      3. fma-neg 76.9%

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

      4. metadata-eval 76.9%

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

   6. Simplified 76.9%

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z - t\right) \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{+252}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 5 \cdot 10^{+262}:\\ \;\;\;\;x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, y \cdot \left(t \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* x_m x_m) 5e+262)
   (+ (* x_m x_m) (* (* y 4.0) (- t (* z z))))
   (fma 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 * x_m) <= 5e+262) {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = fma(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 (Float64(x_m * x_m) <= 5e+262)
		tmp = Float64(Float64(x_m * x_m) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = fma(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[N[(x$95$m * x$95$m), $MachinePrecision], 5e+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[(x$95$m * x$95$m + N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000008e262

   1. Initial program 93.2%

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

   if 5.00000000000000008e262 < (*.f64 x x)

   1. Initial program 72.3%

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

   3. Taylor expanded in z around 0 77.0%

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

      1. associate-*r* 77.0%

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

   5. Simplified 77.0%

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

      1. fma-neg 85.2%

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

      2. *-commutative 85.2%

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

      3. distribute-rgt-neg-in 85.2%

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

      4. distribute-lft-neg-in 85.2%

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

      5. metadata-eval 85.2%

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

   7. Applied egg-rr 85.2%

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

4. Final simplification 91.3%

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

Alternative 4: 92.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 5 \cdot 10^{+265}:\\ \;\;\;\;x\_m \cdot x\_m + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* x_m x_m) 5e+265)
   (+ (* x_m x_m) (* (* y 4.0) (- t (* z z))))
   (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 * x_m) <= 5e+265) {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	} 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 * x_m) <= 5d+265) then
        tmp = (x_m * x_m) + ((y * 4.0d0) * (t - (z * z)))
    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 * x_m) <= 5e+265) {
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	} 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 * x_m) <= 5e+265:
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)))
	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 (Float64(x_m * x_m) <= 5e+265)
		tmp = Float64(Float64(x_m * x_m) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	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 * x_m) <= 5e+265)
		tmp = (x_m * x_m) + ((y * 4.0) * (t - (z * z)));
	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[N[(x$95$m * x$95$m), $MachinePrecision], 5e+265], 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[Power[x$95$m, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.0000000000000002e265

   1. Initial program 93.3%

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

   if 5.0000000000000002e265 < (*.f64 x x)

   1. Initial program 71.8%

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

   3. Taylor expanded in x around 0 73.3%

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

   5. Simplified 84.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+265}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 0.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (let* ((t_1 (+ (* x_m x_m) (* (* y 4.0) (- t (* z z))))))
   (if (<= t_1 INFINITY) t_1 (* -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) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = -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) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = -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) + ((y * 4.0) * (t - (z * z)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = -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(y * 4.0) * Float64(t - Float64(z * z))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 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) + ((y * 4.0) * (t - (z * z)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = -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[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(-4.0 * N[(t * y), $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 94.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 t around inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. *-commutative 0.0%

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

      5. *-commutative 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 12.5%

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

      8. swap-sqr 12.5%

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

      9. metadata-eval 12.5%

         \[\leadsto x \cdot x - \left(\frac{{z}^{2}}{t} \cdot \left(t \cdot \left(y \cdot 4\right)\right) + \left(-1\right) \cdot \left(t \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}}\right)\right) \]

      10. metadata-eval 12.5%

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

      11. swap-sqr 12.5%

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

      12. sqrt-unprod 12.5%

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

      13. add-sqr-sqrt 12.5%

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

      14. associate-*l* 12.5%

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

      15. metadata-eval 12.5%

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

      16. neg-mul-1 12.5%

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

      17. add-sqr-sqrt 12.5%

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

   5. Applied egg-rr 12.5%

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

      1. distribute-lft1-in 25.0%

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

      2. unpow2 25.0%

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

      3. associate-*r/ 25.0%

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

      4. fma-define 25.0%

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

   7. Simplified 25.0%

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

   8. Taylor expanded in x around 0 50.0%

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \left(1 + \frac{{z}^{2}}{t}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 50.0%

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

      2. +-commutative 50.0%

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

      3. unpow2 50.0%

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

      4. associate-*r/ 50.0%

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

      5. fma-undefine 50.0%

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

   10. Simplified 50.0%

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

   11. Taylor expanded in t around inf 44.5%

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

4. Final simplification 91.0%

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

Alternative 6: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{+200}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m - t \cdot \left(y \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 4e+200)
   (- (* x_m x_m) (* y (* t -4.0)))
   (- (* x_m x_m) (* t (* y 4.0)))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 4e+200) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = (x_m * x_m) - (t * (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 (x_m <= 4d+200) then
        tmp = (x_m * x_m) - (y * (t * (-4.0d0)))
    else
        tmp = (x_m * x_m) - (t * (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 (x_m <= 4e+200) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = (x_m * x_m) - (t * (y * 4.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if x_m <= 4e+200:
		tmp = (x_m * x_m) - (y * (t * -4.0))
	else:
		tmp = (x_m * x_m) - (t * (y * 4.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 4e+200)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(x_m * x_m) - Float64(t * 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 (x_m <= 4e+200)
		tmp = (x_m * x_m) - (y * (t * -4.0));
	else
		tmp = (x_m * x_m) - (t * (y * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[x$95$m, 4e+200], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999999e200

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0 63.9%

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

      1. associate-*r* 63.9%

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

   5. Simplified 63.9%

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

   if 3.9999999999999999e200 < x

   1. Initial program 75.0%

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

   3. Taylor expanded in t around inf 75.0%

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

      1. associate-*r* 75.0%

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

      2. sub-neg 75.0%

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

      3. distribute-rgt-in 75.0%

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

      4. *-commutative 75.0%

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

      5. *-commutative 75.0%

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

      6. add-sqr-sqrt 43.8%

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

      7. sqrt-unprod 68.8%

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

      8. swap-sqr 68.8%

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

      9. metadata-eval 68.8%

         \[\leadsto x \cdot x - \left(\frac{{z}^{2}}{t} \cdot \left(t \cdot \left(y \cdot 4\right)\right) + \left(-1\right) \cdot \left(t \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}}\right)\right) \]

      10. metadata-eval 68.8%

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

      11. swap-sqr 68.8%

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

      12. sqrt-unprod 37.5%

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

      13. add-sqr-sqrt 81.3%

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

      14. associate-*l* 81.3%

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

      15. metadata-eval 81.3%

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

      16. neg-mul-1 81.3%

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

      17. add-sqr-sqrt 37.5%

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

   5. Applied egg-rr 81.3%

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

      1. distribute-lft1-in 87.5%

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

      2. unpow2 87.5%

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

      3. associate-*r/ 87.5%

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

      4. fma-define 87.5%

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

   7. Simplified 87.5%

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

   8. Taylor expanded in z around 0 93.8%

      \[\leadsto x \cdot x - \color{blue}{1} \cdot \left(t \cdot \left(y \cdot 4\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+200}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+94}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 2.3e+94) (* 4.0 (* t y)) (* -4.0 (* t y))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 2.3e+94) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = -4.0 * (t * 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 <= 2.3d+94) then
        tmp = 4.0d0 * (t * y)
    else
        tmp = (-4.0d0) * (t * 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 <= 2.3e+94) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = -4.0 * (t * y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if z <= 2.3e+94:
		tmp = 4.0 * (t * y)
	else:
		tmp = -4.0 * (t * y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 2.3e+94)
		tmp = Float64(4.0 * Float64(t * y));
	else
		tmp = Float64(-4.0 * Float64(t * y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if (z <= 2.3e+94)
		tmp = 4.0 * (t * y);
	else
		tmp = -4.0 * (t * y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 2.3e+94], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.3e94

   1. Initial program 91.4%

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

   3. Taylor expanded in t around inf 40.8%

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

      1. *-commutative 40.8%

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

   5. Simplified 40.8%

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

   if 2.3e94 < z

   1. Initial program 70.6%

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

   3. Taylor expanded in t around inf 68.0%

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

      1. associate-*r* 58.2%

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

      2. sub-neg 58.2%

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

      3. distribute-rgt-in 45.4%

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

      4. *-commutative 45.4%

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

      5. *-commutative 45.4%

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

      6. add-sqr-sqrt 11.4%

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

      7. sqrt-unprod 37.7%

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

      8. swap-sqr 37.7%

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

      9. metadata-eval 37.7%

         \[\leadsto x \cdot x - \left(\frac{{z}^{2}}{t} \cdot \left(t \cdot \left(y \cdot 4\right)\right) + \left(-1\right) \cdot \left(t \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}}\right)\right) \]

      10. metadata-eval 37.7%

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

      11. swap-sqr 37.7%

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

      12. sqrt-unprod 31.4%

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

      13. add-sqr-sqrt 50.5%

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

      14. associate-*l* 50.5%

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

      15. metadata-eval 50.5%

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

      16. neg-mul-1 50.5%

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

      17. add-sqr-sqrt 31.4%

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

   5. Applied egg-rr 50.5%

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

      1. distribute-lft1-in 55.6%

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

      2. unpow2 55.6%

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

      3. associate-*r/ 60.4%

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

      4. fma-define 60.4%

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

   7. Simplified 60.4%

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

   8. Taylor expanded in x around 0 65.8%

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \left(1 + \frac{{z}^{2}}{t}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 57.8%

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

      2. +-commutative 57.8%

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

      3. unpow2 57.8%

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

      4. associate-*r/ 62.5%

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

      5. fma-undefine 62.5%

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

   10. Simplified 62.5%

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

   11. Taylor expanded in t around inf 12.1%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+94}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% 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 88.2%

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

3. Taylor expanded in z around 0 65.0%

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

   1. associate-*r* 65.0%

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

5. Simplified 65.0%

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

6. Final simplification 65.0%

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

Alternative 9: 5.8% 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 88.2%

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

3. Taylor expanded in t around inf 85.9%

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

   1. associate-*r* 84.3%

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

   2. sub-neg 84.3%

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

   3. distribute-rgt-in 75.7%

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

   4. *-commutative 75.7%

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

   5. *-commutative 75.7%

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

   6. add-sqr-sqrt 36.8%

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

   7. sqrt-unprod 52.5%

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

   8. swap-sqr 52.5%

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

   9. metadata-eval 52.5%

      \[\leadsto x \cdot x - \left(\frac{{z}^{2}}{t} \cdot \left(t \cdot \left(y \cdot 4\right)\right) + \left(-1\right) \cdot \left(t \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}}\right)\right) \]

   10. metadata-eval 52.5%

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

   11. swap-sqr 52.5%

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

   12. sqrt-unprod 27.7%

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

   13. add-sqr-sqrt 52.8%

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

   14. associate-*l* 52.8%

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

   15. metadata-eval 52.8%

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

   16. neg-mul-1 52.8%

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

   17. add-sqr-sqrt 27.7%

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

5. Applied egg-rr 52.8%

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

   1. distribute-lft1-in 55.2%

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

   2. unpow2 55.2%

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

   3. associate-*r/ 55.9%

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

   4. fma-define 55.9%

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

7. Simplified 55.9%

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

8. Taylor expanded in x around 0 30.3%

   \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \left(1 + \frac{{z}^{2}}{t}\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 30.5%

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

   2. +-commutative 30.5%

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

   3. unpow2 30.5%

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

   4. associate-*r/ 31.2%

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

   5. fma-undefine 31.2%

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

10. Simplified 31.2%

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

11. Taylor expanded in t around inf 8.4%

    \[\leadsto -4 \cdot \color{blue}{\left(t \cdot y\right)} \]
12. Add Preprocessing

Alternative 10: 3.1% accurate, 13.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	return 0.0;
}

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 = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	return 0.0

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, y, z, t)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := 0.0

Derivation

1. Initial program 88.2%

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

3. Taylor expanded in x around 0 63.2%

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

5. Simplified 2.9%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Developer target: 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 2024086 
(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))))