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

Percentage Accurate: 90.7% → 97.2%

Time: 8.2s

Alternatives: 10

Speedup: 0.7×

• 49.3% 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: 97.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \leq \infty:\\ \;\;\;\;x \cdot x - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \frac{y \cdot -4}{\frac{-1}{t}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x x) (* (* y 4.0) (- (* z z) t))) INFINITY)
   (- (* x x) (fma (* -4.0 y) t (* (* (* 4.0 y) z) z)))
   (fma x x (/ (* y -4.0) (/ -1.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) - ((y * 4.0) * ((z * z) - t))) <= ((double) INFINITY)) {
		tmp = (x * x) - fma((-4.0 * y), t, (((4.0 * y) * z) * z));
	} else {
		tmp = fma(x, x, ((y * -4.0) / (-1.0 / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t))) <= Inf)
		tmp = Float64(Float64(x * x) - fma(Float64(-4.0 * y), t, Float64(Float64(Float64(4.0 * y) * z) * z)));
	else
		tmp = fma(x, x, Float64(Float64(y * -4.0) / Float64(-1.0 / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x * x), $MachinePrecision] - N[(N[(-4.0 * y), $MachinePrecision] * t + N[(N[(N[(4.0 * y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(y * -4.0), $MachinePrecision] / N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 95.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 99.9

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 16.7

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 16.7

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

   4. Applied rewrites 16.7%

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

   6. Applied rewrites 0.0%

      \[\leadsto x \cdot x - \color{blue}{\frac{4 \cdot y}{\frac{1}{z \cdot z - t}}} \]

   7. Taylor expanded in z around 0

      \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{\frac{-1}{t}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 66.7

         \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{\frac{-1}{t}}} \]

   9. Applied rewrites 66.7%

      \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{\frac{-1}{t}}} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \color{blue}{x \cdot x - \frac{4 \cdot y}{\frac{-1}{t}}} \]

       2. sub-neg N/A

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

       3. lift-*.f64 N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. distribute-neg-frac N/A

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

       7. lower-/.f64 N/A

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

       8. lift-*.f64 N/A

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

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

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

       10. metadata-eval N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 75.0

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

   11. Applied rewrites 75.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \frac{y \cdot -4}{\frac{-1}{t}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-37}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(-t\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+57}:\\ \;\;\;\;\left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z\right) \cdot -4, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-4 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e-37)
   (- (* x x) (* (* y 4.0) (- t)))
   (if (<= (* z z) 1e+57)
     (* (* (- (* z z) t) y) -4.0)
     (if (<= (* z z) 2e+297)
       (fma (* (* z z) -4.0) y (* x x))
       (* (* z y) (* -4.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e-37) {
		tmp = (x * x) - ((y * 4.0) * -t);
	} else if ((z * z) <= 1e+57) {
		tmp = (((z * z) - t) * y) * -4.0;
	} else if ((z * z) <= 2e+297) {
		tmp = fma(((z * z) * -4.0), y, (x * x));
	} else {
		tmp = (z * y) * (-4.0 * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e-37)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(-t)));
	elseif (Float64(z * z) <= 1e+57)
		tmp = Float64(Float64(Float64(Float64(z * z) - t) * y) * -4.0);
	elseif (Float64(z * z) <= 2e+297)
		tmp = fma(Float64(Float64(z * z) * -4.0), y, Float64(x * x));
	else
		tmp = Float64(Float64(z * y) * Float64(-4.0 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-37], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+57], N[(N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+297], N[(N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(-4.0 * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z z) < 1.00000000000000007e-37

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if 1.00000000000000007e-37 < (*.f64 z z) < 1.00000000000000005e57

   1. Initial program 99.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 94.7

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

   5. Applied rewrites 94.7%

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

   if 1.00000000000000005e57 < (*.f64 z z) < 2e297

   1. Initial program 95.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 95.0

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 85.3

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

   7. Applied rewrites 85.3%

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

   if 2e297 < (*.f64 z z)

   1. Initial program 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 90.5

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 90.5

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

   9. Applied rewrites 86.8%

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

Alternative 3: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-37}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(-t\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+57}:\\ \;\;\;\;\left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-4 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e-37)
   (- (* x x) (* (* y 4.0) (- t)))
   (if (<= (* z z) 1e+57)
     (* (* (- (* z z) t) y) -4.0)
     (if (<= (* z z) 2e+297)
       (fma (* (* z z) y) -4.0 (* x x))
       (* (* z y) (* -4.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e-37) {
		tmp = (x * x) - ((y * 4.0) * -t);
	} else if ((z * z) <= 1e+57) {
		tmp = (((z * z) - t) * y) * -4.0;
	} else if ((z * z) <= 2e+297) {
		tmp = fma(((z * z) * y), -4.0, (x * x));
	} else {
		tmp = (z * y) * (-4.0 * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e-37)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(-t)));
	elseif (Float64(z * z) <= 1e+57)
		tmp = Float64(Float64(Float64(Float64(z * z) - t) * y) * -4.0);
	elseif (Float64(z * z) <= 2e+297)
		tmp = fma(Float64(Float64(z * z) * y), -4.0, Float64(x * x));
	else
		tmp = Float64(Float64(z * y) * Float64(-4.0 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-37], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+57], N[(N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+297], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(-4.0 * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z z) < 1.00000000000000007e-37

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if 1.00000000000000007e-37 < (*.f64 z z) < 1.00000000000000005e57

   1. Initial program 99.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 94.7

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

   5. Applied rewrites 94.7%

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

   if 1.00000000000000005e57 < (*.f64 z z) < 2e297

   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 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if 2e297 < (*.f64 z z)

   1. Initial program 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 90.5

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 90.5

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

   9. Applied rewrites 86.8%

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

Alternative 4: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+297}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \mathsf{fma}\left(z, z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-4 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+297)
   (- (* x x) (* (* y 4.0) (fma z z (- t))))
   (* (* z y) (* -4.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+297) {
		tmp = (x * x) - ((y * 4.0) * fma(z, z, -t));
	} else {
		tmp = (z * y) * (-4.0 * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+297)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * fma(z, z, Float64(-t))));
	else
		tmp = Float64(Float64(z * y) * Float64(-4.0 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+297], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(z * z + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(-4.0 * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e297

   1. Initial program 98.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 98.3

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

   4. Applied rewrites 98.3%

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

   if 2e297 < (*.f64 z z)

   1. Initial program 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 90.5

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 90.5

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

   9. Applied rewrites 86.8%

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

Alternative 5: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+297}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-4 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+297)
   (- (* x x) (* (* y 4.0) (- (* z z) t)))
   (* (* z y) (* -4.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+297) {
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
	} else {
		tmp = (z * y) * (-4.0 * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 2d+297) then
        tmp = (x * x) - ((y * 4.0d0) * ((z * z) - t))
    else
        tmp = (z * y) * ((-4.0d0) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+297) {
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
	} else {
		tmp = (z * y) * (-4.0 * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+297:
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t))
	else:
		tmp = (z * y) * (-4.0 * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+297)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(Float64(z * y) * Float64(-4.0 * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e+297)
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
	else
		tmp = (z * y) * (-4.0 * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+297], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(-4.0 * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e297

   1. Initial program 98.3%

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

   if 2e297 < (*.f64 z z)

   1. Initial program 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 90.5

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 90.5

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

   9. Applied rewrites 86.8%

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

Alternative 6: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{+67}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-4 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 7e+67) (- (* x x) (* (* y 4.0) (- t))) (* (* z y) (* -4.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7e+67) {
		tmp = (x * x) - ((y * 4.0) * -t);
	} else {
		tmp = (z * y) * (-4.0 * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 7d+67) then
        tmp = (x * x) - ((y * 4.0d0) * -t)
    else
        tmp = (z * y) * ((-4.0d0) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7e+67) {
		tmp = (x * x) - ((y * 4.0) * -t);
	} else {
		tmp = (z * y) * (-4.0 * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 7e+67:
		tmp = (x * x) - ((y * 4.0) * -t)
	else:
		tmp = (z * y) * (-4.0 * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 7e+67)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(-t)));
	else
		tmp = Float64(Float64(z * y) * Float64(-4.0 * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 7e+67)
		tmp = (x * x) - ((y * 4.0) * -t);
	else
		tmp = (z * y) * (-4.0 * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 7e+67], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(-4.0 * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7e67

   1. Initial program 96.9%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if 7e67 < z

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 90.0

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

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

   9. Applied rewrites 83.7%

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

Alternative 7: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 3.3 \cdot 10^{+92}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot z\right) \cdot y\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 3.3e+92) (* (* t 4.0) y) (* (* (* z z) y) -4.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 3.3e+92) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = ((z * z) * y) * -4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 3.3d+92) then
        tmp = (t * 4.0d0) * y
    else
        tmp = ((z * z) * y) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 3.3e+92) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = ((z * z) * y) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 3.3e+92:
		tmp = (t * 4.0) * y
	else:
		tmp = ((z * z) * y) * -4.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 3.3e+92)
		tmp = Float64(Float64(t * 4.0) * y);
	else
		tmp = Float64(Float64(Float64(z * z) * y) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 3.3e+92)
		tmp = (t * 4.0) * y;
	else
		tmp = ((z * z) * y) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 3.3e+92], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.29999999999999974e92

   1. Initial program 99.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 56.7

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

   5. Applied rewrites 56.7%

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

   7. Applied rewrites 56.7%

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

   if 3.29999999999999974e92 < (*.f64 z z)

   1. Initial program 80.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

Alternative 8: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-4 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 7e+67) (fma (* t y) 4.0 (* x x)) (* (* z y) (* -4.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7e+67) {
		tmp = fma((t * y), 4.0, (x * x));
	} else {
		tmp = (z * y) * (-4.0 * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 7e+67)
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	else
		tmp = Float64(Float64(z * y) * Float64(-4.0 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 7e+67], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(-4.0 * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7e67

   1. Initial program 96.9%

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

   3. Taylor expanded in 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 77.1

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

   5. Applied rewrites 77.1%

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

   if 7e67 < z

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 90.0

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

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

   9. Applied rewrites 83.7%

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

Alternative 9: 45.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-4 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 2e+50) (* (* t 4.0) y) (* (* z y) (* -4.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2e+50) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = (z * y) * (-4.0 * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 2d+50) then
        tmp = (t * 4.0d0) * y
    else
        tmp = (z * y) * ((-4.0d0) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2e+50) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = (z * y) * (-4.0 * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 2e+50:
		tmp = (t * 4.0) * y
	else:
		tmp = (z * y) * (-4.0 * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2e+50)
		tmp = Float64(Float64(t * 4.0) * y);
	else
		tmp = Float64(Float64(z * y) * Float64(-4.0 * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2e+50)
		tmp = (t * 4.0) * y;
	else
		tmp = (z * y) * (-4.0 * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 2e+50], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(-4.0 * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.0000000000000002e50

   1. Initial program 97.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 46.2%

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

   if 2.0000000000000002e50 < z

   1. Initial program 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 89.2

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 67.4

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

   7. Applied rewrites 67.4%

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

   9. Applied rewrites 79.0%

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

Alternative 10: 32.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
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 38.0

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

5. Applied rewrites 38.0%

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

7. Applied rewrites 38.0%

   \[\leadsto \left(t \cdot 4\right) \cdot \color{blue}{y} \]
8. 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 2024314 
(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))))