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

Percentage Accurate: 90.5% → 96.8%

Time: 10.9s

Alternatives: 10

Speedup: 0.7×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, z, -t\right), y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(z \cdot y\right), -4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+271)
   (fma (fma z z (- t)) (* y -4.0) (* x x))
   (fma (* z (* z y)) -4.0 (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+271) {
		tmp = fma(fma(z, z, -t), (y * -4.0), (x * x));
	} else {
		tmp = fma((z * (z * y)), -4.0, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+271)
		tmp = fma(fma(z, z, Float64(-t)), Float64(y * -4.0), Float64(x * x));
	else
		tmp = fma(Float64(z * Float64(z * y)), -4.0, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+271], N[(N[(z * z + (-t)), $MachinePrecision] * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999991e271

   1. Initial program 99.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lift-*.f64 N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   if 1.99999999999999991e271 < (*.f64 z z)

   1. Initial program 76.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

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

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

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

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. +-commutative N/A

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

   7. Applied rewrites 92.6%

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

Alternative 2: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+235}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, z \cdot -4, t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+41)
   (fma y (* t 4.0) (* x x))
   (if (<= (* z z) 1e+235)
     (* y (fma z (* z -4.0) (* t 4.0)))
     (* -4.0 (* z (* z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+41) {
		tmp = fma(y, (t * 4.0), (x * x));
	} else if ((z * z) <= 1e+235) {
		tmp = y * fma(z, (z * -4.0), (t * 4.0));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+41)
		tmp = fma(y, Float64(t * 4.0), Float64(x * x));
	elseif (Float64(z * z) <= 1e+235)
		tmp = Float64(y * fma(z, Float64(z * -4.0), Float64(t * 4.0)));
	else
		tmp = Float64(-4.0 * Float64(z * Float64(z * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+41], N[(y * N[(t * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+235], N[(y * N[(z * N[(z * -4.0), $MachinePrecision] + N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2.00000000000000001e41

   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. 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 95.9

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

   5. Applied rewrites 95.9%

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

   if 2.00000000000000001e41 < (*.f64 z z) < 1.0000000000000001e235

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-lft-in N/A

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

      13. associate-+l+ N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 68.0

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

   7. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 70.8

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 70.8

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

   9. Applied rewrites 70.8%

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

   if 1.0000000000000001e235 < (*.f64 z z)

   1. Initial program 77.1%

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

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 81.1

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 89.2

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

   7. Applied rewrites 89.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+235}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, z \cdot -4, t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+241}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+41)
   (fma y (* t 4.0) (* x x))
   (if (<= (* z z) 1e+241)
     (* (- (* z z) t) (* y -4.0))
     (* -4.0 (* z (* z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+41) {
		tmp = fma(y, (t * 4.0), (x * x));
	} else if ((z * z) <= 1e+241) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+41)
		tmp = fma(y, Float64(t * 4.0), Float64(x * x));
	elseif (Float64(z * z) <= 1e+241)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0));
	else
		tmp = Float64(-4.0 * Float64(z * Float64(z * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+41], N[(y * N[(t * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+241], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2.00000000000000001e41

   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. 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 95.9

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

   5. Applied rewrites 95.9%

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

   if 2.00000000000000001e41 < (*.f64 z z) < 1.0000000000000001e241

   1. Initial program 99.7%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 70.5

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

   5. Applied rewrites 70.5%

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

   if 1.0000000000000001e241 < (*.f64 z z)

   1. Initial program 76.1%

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

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 81.7

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 90.1

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

   7. Applied rewrites 90.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+241}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((z * z) - t) * (y * 4.0d0)) <= 2d+301) then
        tmp = x * x
    else
        tmp = y * (t * (-4.0d0))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z, t):
	tmp = 0
	if (((z * z) - t) * (y * 4.0)) <= 2e+301:
		tmp = x * x
	else:
		tmp = y * (t * -4.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((((z * z) - t) * (y * 4.0)) <= 2e+301)
		tmp = x * x;
	else
		tmp = y * (t * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)) < 2.00000000000000011e301

   1. Initial program 96.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 51.1

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

   5. Applied rewrites 51.1%

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

   if 2.00000000000000011e301 < (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))

   1. Initial program 77.4%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   7. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\left(\left(y \cdot \mathsf{fma}\left(z, z, t\right)\right) \cdot \left(16 \cdot \left(y \cdot \mathsf{fma}\left(z, z, t\right)\right)\right)\right) \cdot \frac{1}{y \cdot \left(-4 \cdot \mathsf{fma}\left(z, z, t\right)\right)}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot y\right)} \]
   9. 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. lower-*.f64 12.9

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

   10. Applied rewrites 12.9%

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

4. Final simplification 44.7%

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

Alternative 5: 51.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-200}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-97}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+20}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.6e-200)
   (* x x)
   (if (<= z 3.5e-97)
     (* y (* t 4.0))
     (if (<= z 9e+20) (* x x) (* -4.0 (* z (* z y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.6e-200) {
		tmp = x * x;
	} else if (z <= 3.5e-97) {
		tmp = y * (t * 4.0);
	} else if (z <= 9e+20) {
		tmp = x * x;
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3.6d-200) then
        tmp = x * x
    else if (z <= 3.5d-97) then
        tmp = y * (t * 4.0d0)
    else if (z <= 9d+20) then
        tmp = x * x
    else
        tmp = (-4.0d0) * (z * (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.6e-200) {
		tmp = x * x;
	} else if (z <= 3.5e-97) {
		tmp = y * (t * 4.0);
	} else if (z <= 9e+20) {
		tmp = x * x;
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 3.6e-200:
		tmp = x * x
	elif z <= 3.5e-97:
		tmp = y * (t * 4.0)
	elif z <= 9e+20:
		tmp = x * x
	else:
		tmp = -4.0 * (z * (z * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.6e-200)
		tmp = Float64(x * x);
	elseif (z <= 3.5e-97)
		tmp = Float64(y * Float64(t * 4.0));
	elseif (z <= 9e+20)
		tmp = Float64(x * x);
	else
		tmp = Float64(-4.0 * Float64(z * Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.6e-200)
		tmp = x * x;
	elseif (z <= 3.5e-97)
		tmp = y * (t * 4.0);
	elseif (z <= 9e+20)
		tmp = x * x;
	else
		tmp = -4.0 * (z * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 3.6e-200], N[(x * x), $MachinePrecision], If[LessEqual[z, 3.5e-97], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e+20], N[(x * x), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.6000000000000002e-200 or 3.50000000000000019e-97 < z < 9e20

   1. Initial program 96.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 50.9

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

   5. Applied rewrites 50.9%

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

   if 3.6000000000000002e-200 < z < 3.50000000000000019e-97

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 60.6

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

   5. Applied rewrites 60.6%

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

   if 9e20 < z

   1. Initial program 81.1%

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

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 76.0

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.6

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

   7. Applied rewrites 83.6%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-200}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-97}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+20}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-200}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-97}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+20}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.6e-200)
   (* x x)
   (if (<= z 3.5e-97)
     (* y (* t 4.0))
     (if (<= z 9e+20) (* x x) (* -4.0 (* (* z z) y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.6e-200) {
		tmp = x * x;
	} else if (z <= 3.5e-97) {
		tmp = y * (t * 4.0);
	} else if (z <= 9e+20) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3.6d-200) then
        tmp = x * x
    else if (z <= 3.5d-97) then
        tmp = y * (t * 4.0d0)
    else if (z <= 9d+20) then
        tmp = x * x
    else
        tmp = (-4.0d0) * ((z * z) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.6e-200) {
		tmp = x * x;
	} else if (z <= 3.5e-97) {
		tmp = y * (t * 4.0);
	} else if (z <= 9e+20) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 3.6e-200:
		tmp = x * x
	elif z <= 3.5e-97:
		tmp = y * (t * 4.0)
	elif z <= 9e+20:
		tmp = x * x
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.6e-200)
		tmp = Float64(x * x);
	elseif (z <= 3.5e-97)
		tmp = Float64(y * Float64(t * 4.0));
	elseif (z <= 9e+20)
		tmp = Float64(x * x);
	else
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.6e-200)
		tmp = x * x;
	elseif (z <= 3.5e-97)
		tmp = y * (t * 4.0);
	elseif (z <= 9e+20)
		tmp = x * x;
	else
		tmp = -4.0 * ((z * z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 3.6e-200], N[(x * x), $MachinePrecision], If[LessEqual[z, 3.5e-97], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e+20], N[(x * x), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.6000000000000002e-200 or 3.50000000000000019e-97 < z < 9e20

   1. Initial program 96.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 50.9

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

   5. Applied rewrites 50.9%

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

   if 3.6000000000000002e-200 < z < 3.50000000000000019e-97

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 60.6

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

   5. Applied rewrites 60.6%

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

   if 9e20 < z

   1. Initial program 81.1%

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

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-200}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-97}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+20}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999995e38

   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. 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 95.8

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

   5. Applied rewrites 95.8%

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

   if 1.99999999999999995e38 < (*.f64 z z)

   1. Initial program 84.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

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

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

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

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. +-commutative N/A

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

   7. Applied rewrites 89.9%

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

Alternative 8: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+100) (fma y (* t 4.0) (* x x)) (* -4.0 (* z (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+100) {
		tmp = fma(y, (t * 4.0), (x * x));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+100)
		tmp = fma(y, Float64(t * 4.0), Float64(x * x));
	else
		tmp = Float64(-4.0 * Float64(z * Float64(z * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+100], N[(y * N[(t * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000002e100

   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. 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 93.0

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

   5. Applied rewrites 93.0%

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

   if 1.00000000000000002e100 < (*.f64 z z)

   1. Initial program 82.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.1

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 81.2

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

   7. Applied rewrites 81.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.2e-38) (* y (* t 4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.2e-38) {
		tmp = y * (t * 4.0);
	} 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 <= 1.2d-38) then
        tmp = y * (t * 4.0d0)
    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 <= 1.2e-38) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1.2e-38:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.2e-38)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.2e-38)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.2e-38], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.20000000000000011e-38

   1. Initial program 95.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   if 1.20000000000000011e-38 < x

   1. Initial program 86.8%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.3

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

   5. Applied rewrites 65.3%

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

Alternative 10: 40.7% 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 93.2%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 43.1

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

5. Applied rewrites 43.1%

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

Developer Target 1: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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))))