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

Percentage Accurate: 90.7% → 97.4%

Time: 7.8s

Alternatives: 9

Speedup: 0.7×

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.7% 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: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, z, -t\right), y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot y\right) \cdot -4, z, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+218)
   (fma (fma z z (- t)) (* y -4.0) (* x x))
   (fma (* (* z y) -4.0) z (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+218) {
		tmp = fma(fma(z, z, -t), (y * -4.0), (x * x));
	} else {
		tmp = fma(((z * y) * -4.0), z, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+218)
		tmp = fma(fma(z, z, Float64(-t)), Float64(y * -4.0), Float64(x * x));
	else
		tmp = fma(Float64(Float64(z * y) * -4.0), z, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+218], N[(N[(z * z + (-t)), $MachinePrecision] * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * -4.0), $MachinePrecision] * z + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000008e218

   1. Initial program 99.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. *-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 98.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-in N/A

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

      13. lift-neg.f64 N/A

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

      14. sub-neg N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if 1.00000000000000008e218 < (*.f64 z z)

   1. Initial program 71.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 11.8

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

   5. Applied rewrites 11.8%

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

   6. Taylor expanded in t around 0

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

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

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

      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. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 92.6

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

   8. Applied rewrites 92.6%

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

Alternative 2: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, -4 \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot y\right) \cdot -4, z, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+218)
   (fma (- (* z z) t) (* -4.0 y) (* x x))
   (fma (* (* z y) -4.0) z (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+218) {
		tmp = fma(((z * z) - t), (-4.0 * y), (x * x));
	} else {
		tmp = fma(((z * y) * -4.0), z, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+218)
		tmp = fma(Float64(Float64(z * z) - t), Float64(-4.0 * y), Float64(x * x));
	else
		tmp = fma(Float64(Float64(z * y) * -4.0), z, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+218], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * -4.0), $MachinePrecision] * z + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000008e218

   1. Initial program 99.5%

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

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   if 1.00000000000000008e218 < (*.f64 z z)

   1. Initial program 71.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 11.8

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

   5. Applied rewrites 11.8%

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

   6. Taylor expanded in t around 0

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

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

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

      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. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 92.6

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

   8. Applied rewrites 92.6%

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

Alternative 3: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot y\right) \cdot -4, z, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+107)
   (fma (* t y) 4.0 (* x x))
   (fma (* (* z y) -4.0) z (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+107) {
		tmp = fma((t * y), 4.0, (x * x));
	} else {
		tmp = fma(((z * y) * -4.0), z, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+107)
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	else
		tmp = fma(Float64(Float64(z * y) * -4.0), z, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+107], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * -4.0), $MachinePrecision] * z + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e107

   1. Initial program 100.0%

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

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 93.6

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

   5. Applied rewrites 93.6%

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

   if 1.9999999999999999e107 < (*.f64 z z)

   1. Initial program 78.3%

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

   3. Taylor expanded in t around 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 12.2

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in t around 0

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

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

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

      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. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.2

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

   8. Applied rewrites 91.2%

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

Alternative 4: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(z, z, -t\right) \cdot \left(-4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1e-62)
   (* (fma z z (- t)) (* -4.0 y))
   (fma (* t y) 4.0 (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1e-62) {
		tmp = fma(z, z, -t) * (-4.0 * y);
	} else {
		tmp = fma((t * y), 4.0, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1e-62)
		tmp = Float64(fma(z, z, Float64(-t)) * Float64(-4.0 * y));
	else
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-62], N[(N[(z * z + (-t)), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-62

   1. Initial program 95.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 90.7%

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

   if 1e-62 < (*.f64 x x)

   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 z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 89.7

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

   5. Applied rewrites 89.7%

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

Alternative 5: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-62}:\\ \;\;\;\;\left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1e-62)
   (* (* (- (* z z) t) y) -4.0)
   (fma (* t y) 4.0 (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1e-62) {
		tmp = (((z * z) - t) * y) * -4.0;
	} else {
		tmp = fma((t * y), 4.0, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1e-62)
		tmp = Float64(Float64(Float64(Float64(z * z) - t) * y) * -4.0);
	else
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-62], N[(N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-62

   1. Initial program 95.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if 1e-62 < (*.f64 x x)

   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 z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 89.7

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

   5. Applied rewrites 89.7%

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

Alternative 6: 46.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot 4\right) \cdot y\\ \mathbf{if}\;x \leq 1.1 \cdot 10^{-229}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-159}:\\ \;\;\;\;\left(\left(y \cdot z\right) \cdot -4\right) \cdot z\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* t 4.0) y)))
   (if (<= x 1.1e-229)
     t_1
     (if (<= x 7.5e-159)
       (* (* (* y z) -4.0) z)
       (if (<= x 2.6e+26) t_1 (* x x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * 4.0) * y;
	double tmp;
	if (x <= 1.1e-229) {
		tmp = t_1;
	} else if (x <= 7.5e-159) {
		tmp = ((y * z) * -4.0) * z;
	} else if (x <= 2.6e+26) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (t * 4.0d0) * y
    if (x <= 1.1d-229) then
        tmp = t_1
    else if (x <= 7.5d-159) then
        tmp = ((y * z) * (-4.0d0)) * z
    else if (x <= 2.6d+26) then
        tmp = t_1
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * 4.0) * y;
	double tmp;
	if (x <= 1.1e-229) {
		tmp = t_1;
	} else if (x <= 7.5e-159) {
		tmp = ((y * z) * -4.0) * z;
	} else if (x <= 2.6e+26) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * 4.0) * y
	tmp = 0
	if x <= 1.1e-229:
		tmp = t_1
	elif x <= 7.5e-159:
		tmp = ((y * z) * -4.0) * z
	elif x <= 2.6e+26:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * 4.0) * y)
	tmp = 0.0
	if (x <= 1.1e-229)
		tmp = t_1;
	elseif (x <= 7.5e-159)
		tmp = Float64(Float64(Float64(y * z) * -4.0) * z);
	elseif (x <= 2.6e+26)
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * 4.0) * y;
	tmp = 0.0;
	if (x <= 1.1e-229)
		tmp = t_1;
	elseif (x <= 7.5e-159)
		tmp = ((y * z) * -4.0) * z;
	elseif (x <= 2.6e+26)
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[x, 1.1e-229], t$95$1, If[LessEqual[x, 7.5e-159], N[(N[(N[(y * z), $MachinePrecision] * -4.0), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[x, 2.6e+26], t$95$1, N[(x * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.0999999999999999e-229 or 7.5e-159 < x < 2.60000000000000002e26

   1. Initial program 94.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   7. Applied rewrites 42.9%

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

   if 1.0999999999999999e-229 < x < 7.5e-159

   1. Initial program 92.7%

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

   3. Taylor expanded in 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 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 77.7%

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

   if 2.60000000000000002e26 < x

   1. Initial program 84.0%

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

   3. Taylor expanded in z around 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 25.8

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

   5. Applied rewrites 25.8%

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

   7. Applied rewrites 27.2%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.5

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

   10. Applied rewrites 76.5%

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

Alternative 7: 84.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+212) (fma (* t y) 4.0 (* x x)) (* (* (* y z) -4.0) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+212) {
		tmp = fma((t * y), 4.0, (x * x));
	} else {
		tmp = ((y * z) * -4.0) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+212)
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(y * z) * -4.0) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+212], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z), $MachinePrecision] * -4.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999992e212

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if 4.99999999999999992e212 < (*.f64 z z)

   1. Initial program 71.8%

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

   3. Taylor expanded in 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 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 86.7%

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

Alternative 8: 59.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2e+56) (* (* t 4.0) y) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+56) {
		tmp = (t * 4.0) * y;
	} 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 * x) <= 2d+56) then
        tmp = (t * 4.0d0) * y
    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 * x) <= 2e+56) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 2e+56:
		tmp = (t * 4.0) * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2e+56)
		tmp = Float64(Float64(t * 4.0) * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 2e+56)
		tmp = (t * 4.0) * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+56], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000018e56

   1. Initial program 95.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 55.8%

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

   if 2.00000000000000018e56 < (*.f64 x x)

   1. Initial program 88.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 19.2

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

   5. Applied rewrites 19.2%

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

   7. Applied rewrites 20.7%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.7

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

   10. Applied rewrites 80.7%

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

Alternative 9: 40.8% 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 92.0%

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

3. Taylor expanded in z around 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 33.4

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

5. Applied rewrites 33.4%

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

7. Applied rewrites 36.7%

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

8. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 45.5

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

10. Applied rewrites 45.5%

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

Developer Target 1: 90.7% 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 2024307 
(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))))