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

Percentage Accurate: 90.7% → 95.4%

Time: 10.0s

Alternatives: 9

Speedup: 0.6×

• 49.0% 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: 95.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+190}:\\ \;\;\;\;x \cdot x - \mathsf{fma}\left(z \cdot z, y \cdot 4, y \cdot \left(-4 \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \frac{z \cdot \left(y \cdot 4\right)}{\frac{-1}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+190)
   (- (* x x) (fma (* z z) (* y 4.0) (* y (* -4.0 t))))
   (+ (* x x) (/ (* z (* y 4.0)) (/ -1.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+190) {
		tmp = (x * x) - fma((z * z), (y * 4.0), (y * (-4.0 * t)));
	} else {
		tmp = (x * x) + ((z * (y * 4.0)) / (-1.0 / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+190)
		tmp = Float64(Float64(x * x) - fma(Float64(z * z), Float64(y * 4.0), Float64(y * Float64(-4.0 * t))));
	else
		tmp = Float64(Float64(x * x) + Float64(Float64(z * Float64(y * 4.0)) / Float64(-1.0 / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+190], N[(N[(x * x), $MachinePrecision] - N[(N[(z * z), $MachinePrecision] * N[(y * 4.0), $MachinePrecision] + N[(y * N[(-4.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000036e190

   1. Initial program 95.3%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 4\right) \cdot \left(z \cdot z + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{fma}\left(z \cdot z, \color{blue}{y \cdot 4}, \left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)\right)\right)\right) \]

      4. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\left(z \cdot z\right), \color{blue}{\left(y \cdot 4\right)}, \left(\left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{y} \cdot 4\right), \left(\left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, \color{blue}{4}\right), \left(\left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \left(\left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \left(y \cdot \left(4 \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \left(y \cdot \left(\left(\mathsf{neg}\left(t\right)\right) \cdot 4\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(t\right)\right) \cdot 4\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(y, \left(4 \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right) \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(y, \left(4 \cdot \left(-1 \cdot t\right)\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(y, \left(\left(4 \cdot -1\right) \cdot t\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(y, \left(-4 \cdot t\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot t\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(4\right)\right), t\right)\right)\right)\right) \]

      17. metadata-eval 95.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(-4, t\right)\right)\right)\right) \]

   4. Applied egg-rr 95.3%

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

   if 5.00000000000000036e190 < (*.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. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 4\right) \cdot \frac{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}{\color{blue}{z \cdot z + t}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 4\right) \cdot \frac{1}{\color{blue}{\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}}}\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{y \cdot 4}{\color{blue}{\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\left(y \cdot 4\right), \color{blue}{\left(\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\color{blue}{z \cdot z + t}}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{1}{\color{blue}{\frac{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}{z \cdot z + t}}}\right)\right)\right) \]

      7. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{1}{z \cdot z - \color{blue}{t}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \color{blue}{\left(z \cdot z - t\right)}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(z \cdot z\right), \color{blue}{t}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 82.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right)\right)\right)\right) \]

   4. Applied egg-rr 82.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \color{blue}{\left(\frac{1 + \frac{t}{{z}^{2}}}{{z}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{1 + \frac{t}{{z}^{2}}}{z \cdot \color{blue}{z}}\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\frac{1 + \frac{t}{{z}^{2}}}{z}}{\color{blue}{z}}\right)\right)\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\frac{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{t}{{z}^{2}}\right)\right)\right)\right)}{z}}{z}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\frac{1 + \left(\mathsf{neg}\left(-1 \cdot \frac{t}{{z}^{2}}\right)\right)}{z}}{z}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\frac{1 - -1 \cdot \frac{t}{{z}^{2}}}{z}}{z}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\left(\frac{1 - -1 \cdot \frac{t}{{z}^{2}}}{z}\right), \color{blue}{z}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 - -1 \cdot \frac{t}{{z}^{2}}\right), z\right), z\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(-1 \cdot \frac{t}{{z}^{2}}\right)\right)\right), z\right), z\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{t}{{z}^{2}}\right)\right)\right)\right)\right), z\right), z\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{t}{{z}^{2}}\right), z\right), z\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{t}{{z}^{2}}\right)\right), z\right), z\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(t, \left({z}^{2}\right)\right)\right), z\right), z\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), z\right), z\right)\right)\right) \]

      14. *-lowering-*.f64 82.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), z\right), z\right)\right)\right) \]

   7. Simplified 82.9%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{y \cdot 4}{\frac{1 + \frac{t}{z \cdot z}}{z}} \cdot \color{blue}{z}\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\left(y \cdot 4\right) \cdot z}{\color{blue}{\frac{1 + \frac{t}{z \cdot z}}{z}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\left(\left(y \cdot 4\right) \cdot z\right), \color{blue}{\left(\frac{1 + \frac{t}{z \cdot z}}{z}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(y \cdot 4\right), z\right), \left(\frac{\color{blue}{1 + \frac{t}{z \cdot z}}}{z}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \left(\frac{\color{blue}{1} + \frac{t}{z \cdot z}}{z}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \mathsf{/.f64}\left(\left(1 + \frac{t}{z \cdot z}\right), \color{blue}{z}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{t}{z \cdot z}\right)\right), z\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), z\right)\right)\right) \]

      9. *-lowering-*.f64 96.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), z\right)\right)\right) \]

   9. Applied egg-rr 96.4%

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

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \color{blue}{\left(\frac{1}{z}\right)}\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 96.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \mathsf{/.f64}\left(1, \color{blue}{z}\right)\right)\right) \]

   12. Simplified 96.4%

       \[\leadsto x \cdot x - \frac{\left(y \cdot 4\right) \cdot z}{\color{blue}{\frac{1}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+190}:\\ \;\;\;\;x \cdot x - \mathsf{fma}\left(z \cdot z, y \cdot 4, y \cdot \left(-4 \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \frac{z \cdot \left(y \cdot 4\right)}{\frac{-1}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 44.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(4 \cdot t\right)\\ \mathbf{if}\;x \leq 7.8 \cdot 10^{-221}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-50}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* 4.0 t))))
   (if (<= x 7.8e-221)
     t_1
     (if (<= x 5.5e-50)
       (* -4.0 (* (* z z) y))
       (if (<= x 3.5e+21) t_1 (* x x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (4.0 * t);
	double tmp;
	if (x <= 7.8e-221) {
		tmp = t_1;
	} else if (x <= 5.5e-50) {
		tmp = -4.0 * ((z * z) * y);
	} else if (x <= 3.5e+21) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (4.0d0 * t)
    if (x <= 7.8d-221) then
        tmp = t_1
    else if (x <= 5.5d-50) then
        tmp = (-4.0d0) * ((z * z) * y)
    else if (x <= 3.5d+21) then
        tmp = t_1
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (4.0 * t);
	double tmp;
	if (x <= 7.8e-221) {
		tmp = t_1;
	} else if (x <= 5.5e-50) {
		tmp = -4.0 * ((z * z) * y);
	} else if (x <= 3.5e+21) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (4.0 * t)
	tmp = 0
	if x <= 7.8e-221:
		tmp = t_1
	elif x <= 5.5e-50:
		tmp = -4.0 * ((z * z) * y)
	elif x <= 3.5e+21:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(4.0 * t))
	tmp = 0.0
	if (x <= 7.8e-221)
		tmp = t_1;
	elseif (x <= 5.5e-50)
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	elseif (x <= 3.5e+21)
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (4.0 * t);
	tmp = 0.0;
	if (x <= 7.8e-221)
		tmp = t_1;
	elseif (x <= 5.5e-50)
		tmp = -4.0 * ((z * z) * y);
	elseif (x <= 3.5e+21)
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 7.8e-221], t$95$1, If[LessEqual[x, 5.5e-50], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+21], t$95$1, N[(x * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.7999999999999997e-221 or 5.49999999999999975e-50 < x < 3.5e21

   1. Initial program 93.1%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(4 \cdot t\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(t \cdot \color{blue}{4}\right)\right) \]

      5. *-lowering-*.f64 36.4%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(t, \color{blue}{4}\right)\right) \]

   5. Simplified 36.4%

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

   if 7.7999999999999997e-221 < x < 5.49999999999999975e-50

   1. Initial program 91.7%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 53.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 53.4%

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

   if 3.5e21 < 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

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

      2. *-lowering-*.f64 72.5%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 72.5%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-221}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-50}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+190}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \frac{z \cdot \left(y \cdot 4\right)}{\frac{-1}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+190)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (+ (* x x) (/ (* z (* y 4.0)) (/ -1.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+190) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) + ((z * (y * 4.0)) / (-1.0 / 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) <= 5d+190) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) + ((z * (y * 4.0d0)) / ((-1.0d0) / 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) <= 5e+190) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) + ((z * (y * 4.0)) / (-1.0 / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+190:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) + ((z * (y * 4.0)) / (-1.0 / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+190)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) + Float64(Float64(z * Float64(y * 4.0)) / Float64(-1.0 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e+190)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) + ((z * (y * 4.0)) / (-1.0 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+190], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000036e190

   1. Initial program 95.3%

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

   if 5.00000000000000036e190 < (*.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. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 4\right) \cdot \frac{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}{\color{blue}{z \cdot z + t}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 4\right) \cdot \frac{1}{\color{blue}{\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}}}\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{y \cdot 4}{\color{blue}{\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\left(y \cdot 4\right), \color{blue}{\left(\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\color{blue}{z \cdot z + t}}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{1}{\color{blue}{\frac{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}{z \cdot z + t}}}\right)\right)\right) \]

      7. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{1}{z \cdot z - \color{blue}{t}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \color{blue}{\left(z \cdot z - t\right)}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(z \cdot z\right), \color{blue}{t}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 82.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right)\right)\right)\right) \]

   4. Applied egg-rr 82.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \color{blue}{\left(\frac{1 + \frac{t}{{z}^{2}}}{{z}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{1 + \frac{t}{{z}^{2}}}{z \cdot \color{blue}{z}}\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\frac{1 + \frac{t}{{z}^{2}}}{z}}{\color{blue}{z}}\right)\right)\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\frac{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{t}{{z}^{2}}\right)\right)\right)\right)}{z}}{z}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\frac{1 + \left(\mathsf{neg}\left(-1 \cdot \frac{t}{{z}^{2}}\right)\right)}{z}}{z}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\frac{1 - -1 \cdot \frac{t}{{z}^{2}}}{z}}{z}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\left(\frac{1 - -1 \cdot \frac{t}{{z}^{2}}}{z}\right), \color{blue}{z}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 - -1 \cdot \frac{t}{{z}^{2}}\right), z\right), z\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(-1 \cdot \frac{t}{{z}^{2}}\right)\right)\right), z\right), z\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{t}{{z}^{2}}\right)\right)\right)\right)\right), z\right), z\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{t}{{z}^{2}}\right), z\right), z\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{t}{{z}^{2}}\right)\right), z\right), z\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(t, \left({z}^{2}\right)\right)\right), z\right), z\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), z\right), z\right)\right)\right) \]

      14. *-lowering-*.f64 82.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), z\right), z\right)\right)\right) \]

   7. Simplified 82.9%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{y \cdot 4}{\frac{1 + \frac{t}{z \cdot z}}{z}} \cdot \color{blue}{z}\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\left(y \cdot 4\right) \cdot z}{\color{blue}{\frac{1 + \frac{t}{z \cdot z}}{z}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\left(\left(y \cdot 4\right) \cdot z\right), \color{blue}{\left(\frac{1 + \frac{t}{z \cdot z}}{z}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(y \cdot 4\right), z\right), \left(\frac{\color{blue}{1 + \frac{t}{z \cdot z}}}{z}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \left(\frac{\color{blue}{1} + \frac{t}{z \cdot z}}{z}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \mathsf{/.f64}\left(\left(1 + \frac{t}{z \cdot z}\right), \color{blue}{z}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{t}{z \cdot z}\right)\right), z\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), z\right)\right)\right) \]

      9. *-lowering-*.f64 96.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), z\right)\right)\right) \]

   9. Applied egg-rr 96.4%

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

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \color{blue}{\left(\frac{1}{z}\right)}\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 96.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), z\right), \mathsf{/.f64}\left(1, \color{blue}{z}\right)\right)\right) \]

   12. Simplified 96.4%

       \[\leadsto x \cdot x - \frac{\left(y \cdot 4\right) \cdot z}{\color{blue}{\frac{1}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+190}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \frac{z \cdot \left(y \cdot 4\right)}{\frac{-1}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.55 \cdot 10^{+296}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2.55e+296) (+ (* x x) (* (* y 4.0) (- t (* z z)))) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2.55e+296) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 2.55d+296) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2.55e+296) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 2.55e+296:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2.55e+296)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 2.55e+296)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.55e+296], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.5500000000000001e296

   1. Initial program 95.4%

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

   if 2.5500000000000001e296 < (*.f64 x x)

   1. Initial program 80.8%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 91.8%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 91.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.55 \cdot 10^{+296}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-221}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+56}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 8.2e-221)
   (* y (* 4.0 t))
   (if (<= x 2.6e+56) (* z (* z (* y -4.0))) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 8.2e-221) {
		tmp = y * (4.0 * t);
	} else if (x <= 2.6e+56) {
		tmp = z * (z * (y * -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 <= 8.2d-221) then
        tmp = y * (4.0d0 * t)
    else if (x <= 2.6d+56) then
        tmp = z * (z * (y * (-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 <= 8.2e-221) {
		tmp = y * (4.0 * t);
	} else if (x <= 2.6e+56) {
		tmp = z * (z * (y * -4.0));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 8.2e-221:
		tmp = y * (4.0 * t)
	elif x <= 2.6e+56:
		tmp = z * (z * (y * -4.0))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 8.2e-221)
		tmp = Float64(y * Float64(4.0 * t));
	elseif (x <= 2.6e+56)
		tmp = Float64(z * Float64(z * Float64(y * -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 <= 8.2e-221)
		tmp = y * (4.0 * t);
	elseif (x <= 2.6e+56)
		tmp = z * (z * (y * -4.0));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 8.2e-221], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+56], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.19999999999999962e-221

   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 t around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(4 \cdot t\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(t \cdot \color{blue}{4}\right)\right) \]

      5. *-lowering-*.f64 34.3%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(t, \color{blue}{4}\right)\right) \]

   5. Simplified 34.3%

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

   if 8.19999999999999962e-221 < x < 2.60000000000000011e56

   1. Initial program 93.1%

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

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 44.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 44.4%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right), \color{blue}{z}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right), z\right), z\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(4\right)\right)\right), z\right), z\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot -4\right), z\right), z\right) \]

      10. *-lowering-*.f64 51.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, -4\right), z\right), z\right) \]

   7. Applied egg-rr 51.2%

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

   if 2.60000000000000011e56 < x

   1. Initial program 84.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 79.4%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 79.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-221}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+56}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.25 \cdot 10^{+113}:\\ \;\;\;\;-4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2.25e+113) (* -4.0 (* y (- (* z z) t))) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2.25e+113) {
		tmp = -4.0 * (y * ((z * z) - t));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 2.25d+113) then
        tmp = (-4.0d0) * (y * ((z * z) - t))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2.25e+113) {
		tmp = -4.0 * (y * ((z * z) - t));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 2.25e+113:
		tmp = -4.0 * (y * ((z * z) - t))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2.25e+113)
		tmp = Float64(-4.0 * Float64(y * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 2.25e+113)
		tmp = -4.0 * (y * ((z * z) - t));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.25e+113], N[(-4.0 * N[(y * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.25e113

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2} - t\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left({z}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(z \cdot z\right), t\right)\right)\right) \]

      5. *-lowering-*.f64 87.6%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right)\right)\right) \]

   5. Simplified 87.6%

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

   if 2.25e113 < (*.f64 x x)

   1. Initial program 85.4%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 80.6%

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

Alternative 7: 75.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.6 \cdot 10^{+94}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 8.6e+94) (- (* x x) (* -4.0 (* y t))) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8.6e+94) {
		tmp = (x * x) - (-4.0 * (y * t));
	} 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 <= 8.6d+94) then
        tmp = (x * x) - ((-4.0d0) * (y * t))
    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 <= 8.6e+94) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 8.6e+94:
		tmp = (x * x) - (-4.0 * (y * t))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 8.6e+94)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(y * t)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 8.6e+94)
		tmp = (x * x) - (-4.0 * (y * t));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 8.6e+94], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.6e94

   1. Initial program 92.4%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(-4 \cdot \left(t \cdot y\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \color{blue}{\left(t \cdot y\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 74.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   5. Simplified 74.5%

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

   if 8.6e94 < z

   1. Initial program 84.3%

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

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 84.4%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right), \color{blue}{z}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right), z\right), z\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(4\right)\right)\right), z\right), z\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot -4\right), z\right), z\right) \]

      10. *-lowering-*.f64 91.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, -4\right), z\right), z\right) \]

   7. Applied egg-rr 91.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.6 \cdot 10^{+94}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+22}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.05e+22) (* y (* 4.0 t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.05e+22) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 1.05d+22) then
        tmp = y * (4.0d0 * t)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.05e+22) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1.05e+22:
		tmp = y * (4.0 * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.05e+22)
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.05e+22)
		tmp = y * (4.0 * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.05e+22], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0499999999999999e22

   1. Initial program 92.8%

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(4 \cdot t\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(t \cdot \color{blue}{4}\right)\right) \]

      5. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(t, \color{blue}{4}\right)\right) \]

   5. Simplified 36.6%

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

   if 1.0499999999999999e22 < 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

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

      2. *-lowering-*.f64 72.5%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 72.5%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+22}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.3% accurate, 4.3× 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 91.2%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 43.0%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

5. Simplified 43.0%

   \[\leadsto \color{blue}{x \cdot x} \]
6. 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 2024150 
(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))))