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

Percentage Accurate: 91.0% → 98.0%

Time: 8.7s

Alternatives: 8

Speedup: 0.7×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.0% 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: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y \cdot -4\right), z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 \cdot \frac{y}{x \cdot x}, t, \frac{z}{x} \cdot \frac{\left(z \cdot y\right) \cdot -4}{x}\right), x, x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 5e+51)
   (fma (* z (* y -4.0)) z (fma (* (- t) y) -4.0 (* x x)))
   (*
    (fma (fma (* 4.0 (/ y (* x x))) t (* (/ z x) (/ (* (* z y) -4.0) x))) x x)
    x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5e+51) {
		tmp = fma((z * (y * -4.0)), z, fma((-t * y), -4.0, (x * x)));
	} else {
		tmp = fma(fma((4.0 * (y / (x * x))), t, ((z / x) * (((z * y) * -4.0) / x))), x, x) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 5e+51)
		tmp = fma(Float64(z * Float64(y * -4.0)), z, fma(Float64(Float64(-t) * y), -4.0, Float64(x * x)));
	else
		tmp = Float64(fma(fma(Float64(4.0 * Float64(y / Float64(x * x))), t, Float64(Float64(z / x) * Float64(Float64(Float64(z * y) * -4.0) / x))), x, x) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+51], N[(N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-t) * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(4.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t + N[(N[(z / x), $MachinePrecision] * N[(N[(N[(z * y), $MachinePrecision] * -4.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5e51

   1. Initial program 94.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 99.9%

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

   if 5e51 < (*.f64 x x)

   1. Initial program 87.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 95.6%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y \cdot -4\right), z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 \cdot \frac{y}{x \cdot x}, t, \frac{z}{x} \cdot \frac{\left(z \cdot y\right) \cdot -4}{x}\right), x, x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(4 \cdot y\right) \cdot \left(z \cdot z - t\right)\\ t_2 := \mathsf{fma}\left(z \cdot y, z, \left(-t\right) \cdot y\right) \cdot -4\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+243}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* 4.0 y) (- (* z z) t)))
        (t_2 (* (fma (* z y) z (* (- t) y)) -4.0)))
   (if (<= t_1 -2e+243)
     t_2
     (if (<= t_1 5e+273) (fma (* t y) 4.0 (* x x)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (4.0 * y) * ((z * z) - t);
	double t_2 = fma((z * y), z, (-t * y)) * -4.0;
	double tmp;
	if (t_1 <= -2e+243) {
		tmp = t_2;
	} else if (t_1 <= 5e+273) {
		tmp = fma((t * y), 4.0, (x * x));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(4.0 * y) * Float64(Float64(z * z) - t))
	t_2 = Float64(fma(Float64(z * y), z, Float64(Float64(-t) * y)) * -4.0)
	tmp = 0.0
	if (t_1 <= -2e+243)
		tmp = t_2;
	elseif (t_1 <= 5e+273)
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(4.0 * y), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z * y), $MachinePrecision] * z + N[((-t) * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+243], t$95$2, If[LessEqual[t$95$1, 5e+273], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)) < -2.0000000000000001e243 or 4.99999999999999961e273 < (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))

   1. Initial program 79.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 84.1

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 90.3%

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

   if -2.0000000000000001e243 < (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)) < 4.99999999999999961e273

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot \left(z \cdot z - t\right) \leq -2 \cdot 10^{+243}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, z, \left(-t\right) \cdot y\right) \cdot -4\\ \mathbf{elif}\;\left(4 \cdot y\right) \cdot \left(z \cdot z - t\right) \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, z, \left(-t\right) \cdot y\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot -4\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 -5e-17)
     (* (* 4.0 y) t)
     (if (<= t_1 2e+149) (* x x) (* (* (* z -4.0) y) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -5e-17) {
		tmp = (4.0 * y) * t;
	} else if (t_1 <= 2e+149) {
		tmp = x * x;
	} else {
		tmp = ((z * -4.0) * y) * z;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) - t
    if (t_1 <= (-5d-17)) then
        tmp = (4.0d0 * y) * t
    else if (t_1 <= 2d+149) then
        tmp = x * x
    else
        tmp = ((z * (-4.0d0)) * y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -5e-17) {
		tmp = (4.0 * y) * t;
	} else if (t_1 <= 2e+149) {
		tmp = x * x;
	} else {
		tmp = ((z * -4.0) * y) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= -5e-17:
		tmp = (4.0 * y) * t
	elif t_1 <= 2e+149:
		tmp = x * x
	else:
		tmp = ((z * -4.0) * y) * z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (t_1 <= -5e-17)
		tmp = Float64(Float64(4.0 * y) * t);
	elseif (t_1 <= 2e+149)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(Float64(z * -4.0) * y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) - t;
	tmp = 0.0;
	if (t_1 <= -5e-17)
		tmp = (4.0 * y) * t;
	elseif (t_1 <= 2e+149)
		tmp = x * x;
	else
		tmp = ((z * -4.0) * y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-17], N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+149], N[(x * x), $MachinePrecision], N[(N[(N[(z * -4.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -4.9999999999999999e-17

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if -4.9999999999999999e-17 < (-.f64 (*.f64 z z) t) < 2.0000000000000001e149

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   if 2.0000000000000001e149 < (-.f64 (*.f64 z z) t)

   1. Initial program 81.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 72.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;z \cdot z - t \leq 2 \cdot 10^{+149}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot -4\right) \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot z\right) \cdot y\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 -5e-17)
     (* (* 4.0 y) t)
     (if (<= t_1 2e+149) (* x x) (* (* (* z z) y) -4.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -5e-17) {
		tmp = (4.0 * y) * t;
	} else if (t_1 <= 2e+149) {
		tmp = x * x;
	} else {
		tmp = ((z * z) * y) * -4.0;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) - t
    if (t_1 <= (-5d-17)) then
        tmp = (4.0d0 * y) * t
    else if (t_1 <= 2d+149) then
        tmp = x * x
    else
        tmp = ((z * z) * y) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -5e-17) {
		tmp = (4.0 * y) * t;
	} else if (t_1 <= 2e+149) {
		tmp = x * x;
	} else {
		tmp = ((z * z) * y) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= -5e-17:
		tmp = (4.0 * y) * t
	elif t_1 <= 2e+149:
		tmp = x * x
	else:
		tmp = ((z * z) * y) * -4.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (t_1 <= -5e-17)
		tmp = Float64(Float64(4.0 * y) * t);
	elseif (t_1 <= 2e+149)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(Float64(z * z) * y) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) - t;
	tmp = 0.0;
	if (t_1 <= -5e-17)
		tmp = (4.0 * y) * t;
	elseif (t_1 <= 2e+149)
		tmp = x * x;
	else
		tmp = ((z * z) * y) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-17], N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+149], N[(x * x), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -4.9999999999999999e-17

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if -4.9999999999999999e-17 < (-.f64 (*.f64 z z) t) < 2.0000000000000001e149

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   if 2.0000000000000001e149 < (-.f64 (*.f64 z z) t)

   1. Initial program 81.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;z \cdot z - t \leq 2 \cdot 10^{+149}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot z\right) \cdot y\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot -4\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) INFINITY)
   (fma x x (* (* (- (* z z) t) y) -4.0))
   (* (* (* z -4.0) y) z)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < +inf.0

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 94.0

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

   4. Applied rewrites 94.0%

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

   if +inf.0 < (*.f64 z z)

   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 inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.0

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

   5. Applied rewrites 34.0%

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

   7. Applied rewrites 37.6%

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

Alternative 6: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot -4\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+133) (fma (* t y) 4.0 (* x x)) (* (* (* z -4.0) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+133) {
		tmp = fma((t * y), 4.0, (x * x));
	} else {
		tmp = ((z * -4.0) * y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+133)
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(z * -4.0) * y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+133], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * -4.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999961e133

   1. Initial program 97.6%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if 4.99999999999999961e133 < (*.f64 z z)

   1. Initial program 80.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   7. Applied rewrites 80.8%

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

Alternative 7: 46.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-5}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.16e-5) (* (* 4.0 y) t) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.16e-5) {
		tmp = (4.0 * y) * t;
	} else {
		tmp = x * x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= 1.16d-5) then
        tmp = (4.0d0 * y) * t
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.16e-5) {
		tmp = (4.0 * y) * t;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1.16e-5:
		tmp = (4.0 * y) * t
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.16e-5)
		tmp = Float64(Float64(4.0 * y) * t);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.16e-5)
		tmp = (4.0 * y) * t;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.16e-5], N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1600000000000001e-5

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 39.1

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

   5. Applied rewrites 39.1%

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

   if 1.1600000000000001e-5 < x

   1. Initial program 88.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-5}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x * x), $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 y around 0

   \[\leadsto \color{blue}{{x}^{2}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. lower-*.f64 45.3

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

5. Applied rewrites 45.3%

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

Developer Target 1: 91.0% 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 2024256 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

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