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

Percentage Accurate: 90.2% → 94.8%

Time: 7.7s

Alternatives: 7

Speedup: 0.7×

• 50.6% 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: 90.2% 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: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+199}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+199)
   (fma x x (* -4.0 (* y (- (* z z) t))))
   (fma (* (* -4.0 y) z) z (fma (* (- t) y) -4.0 (* x x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+199) {
		tmp = fma(x, x, (-4.0 * (y * ((z * z) - t))));
	} else {
		tmp = fma(((-4.0 * y) * z), z, fma((-t * y), -4.0, (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+199)
		tmp = fma(x, x, Float64(-4.0 * Float64(y * Float64(Float64(z * z) - t))));
	else
		tmp = fma(Float64(Float64(-4.0 * y) * z), z, fma(Float64(Float64(-t) * y), -4.0, Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+199], N[(x * x + N[(-4.0 * N[(y * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * z), $MachinePrecision] * z + N[(N[((-t) * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e199

   1. Initial program 96.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 99.3

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

   4. Applied rewrites 99.3%

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

   if 1.0000000000000001e199 < (*.f64 z z)

   1. Initial program 75.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 94.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+199}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot \left(z \cdot z\right)\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-112)
   (fma (* 4.0 t) y (* x x))
   (if (<= (* z z) 5e+291)
     (fma x x (* (* y (* z z)) -4.0))
     (* (* -4.0 z) (* y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-112) {
		tmp = fma((4.0 * t), y, (x * x));
	} else if ((z * z) <= 5e+291) {
		tmp = fma(x, x, ((y * (z * z)) * -4.0));
	} else {
		tmp = (-4.0 * z) * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e-112)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	elseif (Float64(z * z) <= 5e+291)
		tmp = fma(x, x, Float64(Float64(y * Float64(z * z)) * -4.0));
	else
		tmp = Float64(Float64(-4.0 * z) * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-112], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+291], N[(x * x + N[(N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5.00000000000000044e-112

   1. Initial program 97.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{{x}^{2} - -4 \cdot \left(t \cdot y\right)} \]
   7. 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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 95.7

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

   8. Applied rewrites 95.7%

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

   if 5.00000000000000044e-112 < (*.f64 z z) < 5.0000000000000001e291

   1. Initial program 96.9%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

   7. Applied rewrites 90.8%

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

   if 5.0000000000000001e291 < (*.f64 z z)

   1. Initial program 68.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   7. Applied rewrites 85.5%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot \left(z \cdot z\right)\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot z, -4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+212)
   (fma x x (* -4.0 (* y (- (* z z) t))))
   (fma (* (* y z) z) -4.0 (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+212) {
		tmp = fma(x, x, (-4.0 * (y * ((z * z) - t))));
	} else {
		tmp = fma(((y * z) * z), -4.0, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+212)
		tmp = fma(x, x, Float64(-4.0 * Float64(y * Float64(Float64(z * z) - t))));
	else
		tmp = fma(Float64(Float64(y * z) * z), -4.0, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+212], N[(x * x + N[(-4.0 * N[(y * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999998e212

   1. Initial program 97.0%

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

      1. 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 99.3

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

   4. Applied rewrites 99.3%

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

   if 1.9999999999999998e212 < (*.f64 z z)

   1. Initial program 73.8%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 94.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot z, -4, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot z, -4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-112)
   (fma (* 4.0 t) y (* x x))
   (fma (* (* y z) z) -4.0 (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-112) {
		tmp = fma((4.0 * t), y, (x * x));
	} else {
		tmp = fma(((y * z) * z), -4.0, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e-112)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	else
		tmp = fma(Float64(Float64(y * z) * z), -4.0, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-112], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000044e-112

   1. Initial program 97.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{{x}^{2} - -4 \cdot \left(t \cdot y\right)} \]
   7. 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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 95.7

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

   8. Applied rewrites 95.7%

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

   if 5.00000000000000044e-112 < (*.f64 z z)

   1. Initial program 82.3%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 91.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot z, -4, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-8}:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e-8) (* (* 4.0 t) y) (* (* -4.0 z) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e-8) {
		tmp = (4.0 * t) * y;
	} else {
		tmp = (-4.0 * z) * (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) :: tmp
    if ((z * z) <= 1d-8) then
        tmp = (4.0d0 * t) * y
    else
        tmp = ((-4.0d0) * z) * (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e-8) {
		tmp = (4.0 * t) * y;
	} else {
		tmp = (-4.0 * z) * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e-8:
		tmp = (4.0 * t) * y
	else:
		tmp = (-4.0 * z) * (y * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e-8)
		tmp = Float64(Float64(4.0 * t) * y);
	else
		tmp = Float64(Float64(-4.0 * z) * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e-8)
		tmp = (4.0 * t) * y;
	else
		tmp = (-4.0 * z) * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-8], N[(N[(4.0 * t), $MachinePrecision] * y), $MachinePrecision], N[(N[(-4.0 * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-8

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 49.3

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

   5. Applied rewrites 49.3%

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

   7. Applied rewrites 49.3%

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

   if 1e-8 < (*.f64 z z)

   1. Initial program 80.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 60.8

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

   5. Applied rewrites 60.8%

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

   7. Applied rewrites 69.5%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-8}:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.9e+81) (fma (* 4.0 t) y (* x x)) (* (* -4.0 z) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.9e+81) {
		tmp = fma((4.0 * t), y, (x * x));
	} else {
		tmp = (-4.0 * z) * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.9e+81)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	else
		tmp = Float64(Float64(-4.0 * z) * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.9e+81], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.9e81

   1. Initial program 91.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 33.8

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{{x}^{2} - -4 \cdot \left(t \cdot y\right)} \]
   7. 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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.5

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

   8. Applied rewrites 74.5%

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

   if 1.9e81 < z

   1. Initial program 79.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 75.6%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

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 = (4.0d0 * t) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.0%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 29.0

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

5. Applied rewrites 29.0%

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

7. Applied rewrites 29.0%

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

8. Final simplification 29.0%

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

Developer Target 1: 90.2% 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 2024327 
(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))))