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

Percentage Accurate: 91.1% → 97.1%

Time: 8.6s

Alternatives: 9

Speedup: 0.8×

• 49.5% 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: 91.1% 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: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot -4\right) \cdot z, z, \mathsf{fma}\left(-4, t \cdot \left(-y\right), x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \mathsf{fma}\left(z, z, -t\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y 4.0) 2e+38)
   (fma (* (* y -4.0) z) z (fma -4.0 (* t (- y)) (* x x)))
   (fma x x (* y (* -4.0 (fma z z (- t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * 4.0) <= 2e+38) {
		tmp = fma(((y * -4.0) * z), z, fma(-4.0, (t * -y), (x * x)));
	} else {
		tmp = fma(x, x, (y * (-4.0 * fma(z, z, -t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * 4.0) <= 2e+38)
		tmp = fma(Float64(Float64(y * -4.0) * z), z, fma(-4.0, Float64(t * Float64(-y)), Float64(x * x)));
	else
		tmp = fma(x, x, Float64(y * Float64(-4.0 * fma(z, z, Float64(-t)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * 4.0), $MachinePrecision], 2e+38], N[(N[(N[(y * -4.0), $MachinePrecision] * z), $MachinePrecision] * z + N[(-4.0 * N[(t * (-y)), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * N[(-4.0 * N[(z * z + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 4 binary64)) < 1.99999999999999995e38

   1. Initial program 85.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. distribute-rgt-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 97.3%

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

   if 1.99999999999999995e38 < (*.f64 y #s(literal 4 binary64))

   1. Initial program 81.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. metadata-eval 95.4

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

   4. Applied rewrites 95.4%

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

4. Final simplification 96.8%

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

Alternative 2: 90.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(y, -4 \cdot \left(z \cdot z\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+64)
   (fma y (* 4.0 t) (* x x))
   (if (<= (* z z) 2e+301)
     (fma y (* -4.0 (* z z)) (* x x))
     (* z (* (* y -4.0) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+64) {
		tmp = fma(y, (4.0 * t), (x * x));
	} else if ((z * z) <= 2e+301) {
		tmp = fma(y, (-4.0 * (z * z)), (x * x));
	} else {
		tmp = z * ((y * -4.0) * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+64)
		tmp = fma(y, Float64(4.0 * t), Float64(x * x));
	elseif (Float64(z * z) <= 2e+301)
		tmp = fma(y, Float64(-4.0 * Float64(z * z)), Float64(x * x));
	else
		tmp = Float64(z * Float64(Float64(y * -4.0) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+64], N[(y * N[(4.0 * t), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+301], N[(y * N[(-4.0 * N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y * -4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.00000000000000002e64

   1. Initial program 97.7%

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

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if 1.00000000000000002e64 < (*.f64 z z) < 2.00000000000000011e301

   1. Initial program 94.0%

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

   3. Taylor expanded in t around 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. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if 2.00000000000000011e301 < (*.f64 z z)

   1. Initial program 54.6%

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   7. Applied rewrites 89.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(y, -4 \cdot \left(z \cdot z\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(4 \cdot t\right)\\ \mathbf{if}\;z \leq 3.5 \cdot 10^{-264}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-205}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* 4.0 t))))
   (if (<= z 3.5e-264)
     t_1
     (if (<= z 2.2e-205)
       (* x x)
       (if (<= z 3.4e-16)
         t_1
         (if (<= z 2.4e+37) (* x x) (* z (* (* y -4.0) z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (4.0 * t);
	double tmp;
	if (z <= 3.5e-264) {
		tmp = t_1;
	} else if (z <= 2.2e-205) {
		tmp = x * x;
	} else if (z <= 3.4e-16) {
		tmp = t_1;
	} else if (z <= 2.4e+37) {
		tmp = x * x;
	} else {
		tmp = z * ((y * -4.0) * 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 = y * (4.0d0 * t)
    if (z <= 3.5d-264) then
        tmp = t_1
    else if (z <= 2.2d-205) then
        tmp = x * x
    else if (z <= 3.4d-16) then
        tmp = t_1
    else if (z <= 2.4d+37) then
        tmp = x * x
    else
        tmp = z * ((y * (-4.0d0)) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (4.0 * t);
	double tmp;
	if (z <= 3.5e-264) {
		tmp = t_1;
	} else if (z <= 2.2e-205) {
		tmp = x * x;
	} else if (z <= 3.4e-16) {
		tmp = t_1;
	} else if (z <= 2.4e+37) {
		tmp = x * x;
	} else {
		tmp = z * ((y * -4.0) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (4.0 * t)
	tmp = 0
	if z <= 3.5e-264:
		tmp = t_1
	elif z <= 2.2e-205:
		tmp = x * x
	elif z <= 3.4e-16:
		tmp = t_1
	elif z <= 2.4e+37:
		tmp = x * x
	else:
		tmp = z * ((y * -4.0) * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(4.0 * t))
	tmp = 0.0
	if (z <= 3.5e-264)
		tmp = t_1;
	elseif (z <= 2.2e-205)
		tmp = Float64(x * x);
	elseif (z <= 3.4e-16)
		tmp = t_1;
	elseif (z <= 2.4e+37)
		tmp = Float64(x * x);
	else
		tmp = Float64(z * Float64(Float64(y * -4.0) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (4.0 * t);
	tmp = 0.0;
	if (z <= 3.5e-264)
		tmp = t_1;
	elseif (z <= 2.2e-205)
		tmp = x * x;
	elseif (z <= 3.4e-16)
		tmp = t_1;
	elseif (z <= 2.4e+37)
		tmp = x * x;
	else
		tmp = z * ((y * -4.0) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 3.5e-264], t$95$1, If[LessEqual[z, 2.2e-205], N[(x * x), $MachinePrecision], If[LessEqual[z, 3.4e-16], t$95$1, If[LessEqual[z, 2.4e+37], N[(x * x), $MachinePrecision], N[(z * N[(N[(y * -4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.5e-264 or 2.20000000000000009e-205 < z < 3.4e-16

   1. Initial program 91.6%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 42.7

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

   5. Applied rewrites 42.7%

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

   if 3.5e-264 < z < 2.20000000000000009e-205 or 3.4e-16 < z < 2.4e37

   1. Initial program 90.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if 2.4e37 < z

   1. Initial program 63.4%

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

   7. Applied rewrites 74.0%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-264}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-205}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(4 \cdot t\right)\\ \mathbf{if}\;z \leq 3.5 \cdot 10^{-264}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-205}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-4 \cdot \left(z \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* 4.0 t))))
   (if (<= z 3.5e-264)
     t_1
     (if (<= z 2.2e-205)
       (* x x)
       (if (<= z 3.4e-16)
         t_1
         (if (<= z 2.4e+37) (* x x) (* y (* -4.0 (* z z)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (4.0 * t);
	double tmp;
	if (z <= 3.5e-264) {
		tmp = t_1;
	} else if (z <= 2.2e-205) {
		tmp = x * x;
	} else if (z <= 3.4e-16) {
		tmp = t_1;
	} else if (z <= 2.4e+37) {
		tmp = x * x;
	} else {
		tmp = y * (-4.0 * (z * 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 = y * (4.0d0 * t)
    if (z <= 3.5d-264) then
        tmp = t_1
    else if (z <= 2.2d-205) then
        tmp = x * x
    else if (z <= 3.4d-16) then
        tmp = t_1
    else if (z <= 2.4d+37) then
        tmp = x * x
    else
        tmp = y * ((-4.0d0) * (z * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (4.0 * t);
	double tmp;
	if (z <= 3.5e-264) {
		tmp = t_1;
	} else if (z <= 2.2e-205) {
		tmp = x * x;
	} else if (z <= 3.4e-16) {
		tmp = t_1;
	} else if (z <= 2.4e+37) {
		tmp = x * x;
	} else {
		tmp = y * (-4.0 * (z * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (4.0 * t)
	tmp = 0
	if z <= 3.5e-264:
		tmp = t_1
	elif z <= 2.2e-205:
		tmp = x * x
	elif z <= 3.4e-16:
		tmp = t_1
	elif z <= 2.4e+37:
		tmp = x * x
	else:
		tmp = y * (-4.0 * (z * z))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(4.0 * t))
	tmp = 0.0
	if (z <= 3.5e-264)
		tmp = t_1;
	elseif (z <= 2.2e-205)
		tmp = Float64(x * x);
	elseif (z <= 3.4e-16)
		tmp = t_1;
	elseif (z <= 2.4e+37)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(-4.0 * Float64(z * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (4.0 * t);
	tmp = 0.0;
	if (z <= 3.5e-264)
		tmp = t_1;
	elseif (z <= 2.2e-205)
		tmp = x * x;
	elseif (z <= 3.4e-16)
		tmp = t_1;
	elseif (z <= 2.4e+37)
		tmp = x * x;
	else
		tmp = y * (-4.0 * (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 3.5e-264], t$95$1, If[LessEqual[z, 2.2e-205], N[(x * x), $MachinePrecision], If[LessEqual[z, 3.4e-16], t$95$1, If[LessEqual[z, 2.4e+37], N[(x * x), $MachinePrecision], N[(y * N[(-4.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.5e-264 or 2.20000000000000009e-205 < z < 3.4e-16

   1. Initial program 91.6%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 42.7

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

   5. Applied rewrites 42.7%

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

   if 3.5e-264 < z < 2.20000000000000009e-205 or 3.4e-16 < z < 2.4e37

   1. Initial program 90.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if 2.4e37 < z

   1. Initial program 63.4%

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-264}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-205}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-4 \cdot \left(z \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \mathsf{fma}\left(z, z, -t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot -4\right) \cdot z, z, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+196)
   (fma x x (* y (* -4.0 (fma z z (- t)))))
   (fma (* (* y -4.0) z) z (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+196) {
		tmp = fma(x, x, (y * (-4.0 * fma(z, z, -t))));
	} else {
		tmp = fma(((y * -4.0) * z), z, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+196)
		tmp = fma(x, x, Float64(y * Float64(-4.0 * fma(z, z, Float64(-t)))));
	else
		tmp = fma(Float64(Float64(y * -4.0) * z), z, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+196], N[(x * x + N[(y * N[(-4.0 * N[(z * z + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * -4.0), $MachinePrecision] * z), $MachinePrecision] * z + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e196

   1. Initial program 98.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. metadata-eval 98.7

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

   4. Applied rewrites 98.7%

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

   if 1.9999999999999999e196 < (*.f64 z z)

   1. Initial program 64.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. distribute-rgt-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 86.3%

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

   5. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 91.1

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

   7. Applied rewrites 91.1%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \mathsf{fma}\left(z, z, -t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot -4\right) \cdot z, z, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot -4\right) \cdot z, z, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+64)
   (fma y (* 4.0 t) (* x x))
   (fma (* (* y -4.0) z) z (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+64) {
		tmp = fma(y, (4.0 * t), (x * x));
	} else {
		tmp = fma(((y * -4.0) * z), z, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+64)
		tmp = fma(y, Float64(4.0 * t), Float64(x * x));
	else
		tmp = fma(Float64(Float64(y * -4.0) * z), z, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+64], N[(y * N[(4.0 * t), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * -4.0), $MachinePrecision] * z), $MachinePrecision] * z + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000002e64

   1. Initial program 97.7%

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

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if 1.00000000000000002e64 < (*.f64 z z)

   1. Initial program 70.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. distribute-rgt-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot -4\right) \cdot z, z, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4e+172) (fma y (* 4.0 t) (* x x)) (* z (* (* y -4.0) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+172) {
		tmp = fma(y, (4.0 * t), (x * x));
	} else {
		tmp = z * ((y * -4.0) * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4e+172)
		tmp = fma(y, Float64(4.0 * t), Float64(x * x));
	else
		tmp = Float64(z * Float64(Float64(y * -4.0) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+172], N[(y * N[(4.0 * t), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y * -4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.0000000000000003e172

   1. Initial program 98.0%

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

   3. Taylor expanded in z around 0

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

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 89.7

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

   5. Applied rewrites 89.7%

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

   if 4.0000000000000003e172 < (*.f64 z z)

   1. Initial program 65.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 84.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.62 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.62e+72) (* y (* 4.0 t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.62e+72) {
		tmp = y * (4.0 * 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 <= 1.62d+72) then
        tmp = y * (4.0d0 * 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 <= 1.62e+72) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1.62e+72:
		tmp = y * (4.0 * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.62e+72)
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.62e+72)
		tmp = y * (4.0 * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.62e+72], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.62000000000000008e72

   1. Initial program 85.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 39.2

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

   5. Applied rewrites 39.2%

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

   if 1.62000000000000008e72 < x

   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 x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.62 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.1% 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 84.5%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 36.6

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

5. Applied rewrites 36.6%

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

Developer Target 1: 91.2% 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 2024223 
(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))))