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

Percentage Accurate: 90.2% → 94.6%

Time: 12.1s

Alternatives: 11

Speedup: 0.8×

• 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: 90.2% 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: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 4.2 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z\_m, z\_m, 0 - t\right), y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \mathsf{fma}\left(z\_m \cdot \left(y \cdot 4\right), z\_m, t \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 4.2e+164)
   (fma (fma z_m z_m (- 0.0 t)) (* y -4.0) (* x x))
   (- (* x x) (fma (* z_m (* y 4.0)) z_m (* t (* y -4.0))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 4.2e+164) {
		tmp = fma(fma(z_m, z_m, (0.0 - t)), (y * -4.0), (x * x));
	} else {
		tmp = (x * x) - fma((z_m * (y * 4.0)), z_m, (t * (y * -4.0)));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 4.2e+164)
		tmp = fma(fma(z_m, z_m, Float64(0.0 - t)), Float64(y * -4.0), Float64(x * x));
	else
		tmp = Float64(Float64(x * x) - fma(Float64(z_m * Float64(y * 4.0)), z_m, Float64(t * Float64(y * -4.0))));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 4.2e+164], N[(N[(z$95$m * z$95$m + N[(0.0 - t), $MachinePrecision]), $MachinePrecision] * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(z$95$m * N[(y * 4.0), $MachinePrecision]), $MachinePrecision] * z$95$m + N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.1999999999999998e164

   1. Initial program 94.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

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

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, \color{blue}{0 - t}\right), \mathsf{neg}\left(y \cdot 4\right), x \cdot x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, \color{blue}{0 - t}\right), \mathsf{neg}\left(y \cdot 4\right), x \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), \color{blue}{y \cdot \left(\mathsf{neg}\left(4\right)\right)}, x \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), \color{blue}{y \cdot \left(\mathsf{neg}\left(4\right)\right)}, x \cdot x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot \color{blue}{-4}, x \cdot x\right) \]

      13. *-lowering-*.f64 95.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, \color{blue}{x \cdot x}\right) \]

   4. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, x \cdot x\right)} \]

   if 4.1999999999999998e164 < z

   1. Initial program 61.9%

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. neg-mul-1 N/A

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

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

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

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

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

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

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

      16. metadata-eval 89.8

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

   4. Applied egg-rr 89.8%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \mathsf{fma}\left(z \cdot \left(y \cdot 4\right), z, t \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{elif}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(z\_m \cdot z\_m, y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 5e+86)
   (fma y (* t 4.0) (* x x))
   (if (<= (* z_m z_m) 2e+287)
     (fma (* z_m z_m) (* y -4.0) (* x x))
     (* z_m (* z_m (* y -4.0))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 5e+86) {
		tmp = fma(y, (t * 4.0), (x * x));
	} else if ((z_m * z_m) <= 2e+287) {
		tmp = fma((z_m * z_m), (y * -4.0), (x * x));
	} else {
		tmp = z_m * (z_m * (y * -4.0));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5e+86)
		tmp = fma(y, Float64(t * 4.0), Float64(x * x));
	elseif (Float64(z_m * z_m) <= 2e+287)
		tmp = fma(Float64(z_m * z_m), Float64(y * -4.0), Float64(x * x));
	else
		tmp = Float64(z_m * Float64(z_m * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e+86], N[(y * N[(t * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+287], N[(N[(z$95$m * z$95$m), $MachinePrecision] * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.9999999999999998e86

   1. Initial program 97.9%

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(y, t \cdot 4, \color{blue}{{x}^{2} + 0}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, t \cdot 4, \color{blue}{x \cdot x} + 0\right) \]

      11. accelerator-lowering-fma.f64 93.5

          \[\leadsto \mathsf{fma}\left(y, t \cdot 4, \color{blue}{\mathsf{fma}\left(x, x, 0\right)}\right) \]

   5. Simplified 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot 4, \mathsf{fma}\left(x, x, 0\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 93.5

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

   7. Applied egg-rr 93.5%

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

   if 4.9999999999999998e86 < (*.f64 z z) < 2.0000000000000002e287

   1. Initial program 97.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

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

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, \color{blue}{0 - t}\right), \mathsf{neg}\left(y \cdot 4\right), x \cdot x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, \color{blue}{0 - t}\right), \mathsf{neg}\left(y \cdot 4\right), x \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), \color{blue}{y \cdot \left(\mathsf{neg}\left(4\right)\right)}, x \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), \color{blue}{y \cdot \left(\mathsf{neg}\left(4\right)\right)}, x \cdot x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot \color{blue}{-4}, x \cdot x\right) \]

      13. *-lowering-*.f64 97.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, \color{blue}{x \cdot x}\right) \]

   4. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, x \cdot x\right)} \]

   5. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{z}^{2} + 0}, y \cdot -4, x \cdot x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z} + 0, y \cdot -4, x \cdot x\right) \]

      3. accelerator-lowering-fma.f64 92.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z, 0\right)}, y \cdot -4, x \cdot x\right) \]

   7. Simplified 92.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z, 0\right)}, y \cdot -4, x \cdot x\right) \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 92.4

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

   9. Applied egg-rr 92.4%

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

   if 2.0000000000000002e287 < (*.f64 z z)

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot {z}^{2}, 0\right)} \]

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

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{y \cdot {z}^{2}}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\left({z}^{2} + 0\right)}, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \left(\color{blue}{z \cdot z} + 0\right), 0\right) \]

      6. accelerator-lowering-fma.f64 74.6

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 0\right) \]

   5. Simplified 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot \mathsf{fma}\left(z, z, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot z + 0\right)\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 84.9

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

   7. Applied egg-rr 84.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z, y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := z\_m \cdot z\_m - t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+143}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (- (* z_m z_m) t)))
   (if (<= t_1 -2e-107)
     (* t (* y 4.0))
     (if (<= t_1 1e+143) (* x x) (* z_m (* z_m (* y -4.0)))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) - t;
	double tmp;
	if (t_1 <= -2e-107) {
		tmp = t * (y * 4.0);
	} else if (t_1 <= 1e+143) {
		tmp = x * x;
	} else {
		tmp = z_m * (z_m * (y * -4.0));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z_m * z_m) - t
    if (t_1 <= (-2d-107)) then
        tmp = t * (y * 4.0d0)
    else if (t_1 <= 1d+143) then
        tmp = x * x
    else
        tmp = z_m * (z_m * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) - t;
	double tmp;
	if (t_1 <= -2e-107) {
		tmp = t * (y * 4.0);
	} else if (t_1 <= 1e+143) {
		tmp = x * x;
	} else {
		tmp = z_m * (z_m * (y * -4.0));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (z_m * z_m) - t
	tmp = 0
	if t_1 <= -2e-107:
		tmp = t * (y * 4.0)
	elif t_1 <= 1e+143:
		tmp = x * x
	else:
		tmp = z_m * (z_m * (y * -4.0))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(z_m * z_m) - t)
	tmp = 0.0
	if (t_1 <= -2e-107)
		tmp = Float64(t * Float64(y * 4.0));
	elseif (t_1 <= 1e+143)
		tmp = Float64(x * x);
	else
		tmp = Float64(z_m * Float64(z_m * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (z_m * z_m) - t;
	tmp = 0.0;
	if (t_1 <= -2e-107)
		tmp = t * (y * 4.0);
	elseif (t_1 <= 1e+143)
		tmp = x * x;
	else
		tmp = z_m * (z_m * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-107], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+143], N[(x * x), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -2e-107

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right) + 0} \]

      2. *-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(y \cdot t\right)} + 0 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(4 \cdot y\right) \cdot t} + 0 \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot 4\right)} \cdot t + 0 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(4 \cdot t\right)} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 4 \cdot t, 0\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, 0\right) \]

      8. *-lowering-*.f64 72.8

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, 0\right) \]

   5. Simplified 72.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot 4, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 72.8

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

   7. Applied egg-rr 72.8%

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

   if -2e-107 < (-.f64 (*.f64 z z) t) < 1e143

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 60.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 60.1

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

   7. Applied egg-rr 60.1%

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

   if 1e143 < (-.f64 (*.f64 z z) t)

   1. Initial program 80.4%

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

   3. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot {z}^{2}, 0\right)} \]

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

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{y \cdot {z}^{2}}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\left({z}^{2} + 0\right)}, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \left(\color{blue}{z \cdot z} + 0\right), 0\right) \]

      6. accelerator-lowering-fma.f64 62.6

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 0\right) \]

   5. Simplified 62.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot \mathsf{fma}\left(z, z, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot z + 0\right)\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 68.5

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

   7. Applied egg-rr 68.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -2 \cdot 10^{-107}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \cdot z - t \leq 10^{+143}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := z\_m \cdot z\_m - t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+143}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z\_m \cdot \left(z\_m \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (- (* z_m z_m) t)))
   (if (<= t_1 -2e-107)
     (* t (* y 4.0))
     (if (<= t_1 1e+143) (* x x) (* y (* z_m (* z_m -4.0)))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) - t;
	double tmp;
	if (t_1 <= -2e-107) {
		tmp = t * (y * 4.0);
	} else if (t_1 <= 1e+143) {
		tmp = x * x;
	} else {
		tmp = y * (z_m * (z_m * -4.0));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z_m * z_m) - t
    if (t_1 <= (-2d-107)) then
        tmp = t * (y * 4.0d0)
    else if (t_1 <= 1d+143) then
        tmp = x * x
    else
        tmp = y * (z_m * (z_m * (-4.0d0)))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) - t;
	double tmp;
	if (t_1 <= -2e-107) {
		tmp = t * (y * 4.0);
	} else if (t_1 <= 1e+143) {
		tmp = x * x;
	} else {
		tmp = y * (z_m * (z_m * -4.0));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (z_m * z_m) - t
	tmp = 0
	if t_1 <= -2e-107:
		tmp = t * (y * 4.0)
	elif t_1 <= 1e+143:
		tmp = x * x
	else:
		tmp = y * (z_m * (z_m * -4.0))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(z_m * z_m) - t)
	tmp = 0.0
	if (t_1 <= -2e-107)
		tmp = Float64(t * Float64(y * 4.0));
	elseif (t_1 <= 1e+143)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(z_m * Float64(z_m * -4.0)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (z_m * z_m) - t;
	tmp = 0.0;
	if (t_1 <= -2e-107)
		tmp = t * (y * 4.0);
	elseif (t_1 <= 1e+143)
		tmp = x * x;
	else
		tmp = y * (z_m * (z_m * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-107], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+143], N[(x * x), $MachinePrecision], N[(y * N[(z$95$m * N[(z$95$m * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -2e-107

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right) + 0} \]

      2. *-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(y \cdot t\right)} + 0 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(4 \cdot y\right) \cdot t} + 0 \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot 4\right)} \cdot t + 0 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(4 \cdot t\right)} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 4 \cdot t, 0\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, 0\right) \]

      8. *-lowering-*.f64 72.8

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, 0\right) \]

   5. Simplified 72.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot 4, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 72.8

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

   7. Applied egg-rr 72.8%

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

   if -2e-107 < (-.f64 (*.f64 z z) t) < 1e143

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 60.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 60.1

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

   7. Applied egg-rr 60.1%

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

   if 1e143 < (-.f64 (*.f64 z z) t)

   1. Initial program 80.4%

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

   3. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot {z}^{2}, 0\right)} \]

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

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{y \cdot {z}^{2}}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\left({z}^{2} + 0\right)}, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \left(\color{blue}{z \cdot z} + 0\right), 0\right) \]

      6. accelerator-lowering-fma.f64 62.6

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 0\right) \]

   5. Simplified 62.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot \mathsf{fma}\left(z, z, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot z + 0\right)\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 68.5

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

   7. Applied egg-rr 68.5%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 62.6

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

   9. Applied egg-rr 62.6%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -2 \cdot 10^{-107}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \cdot z - t \leq 10^{+143}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(z \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z\_m, z\_m, 0 - t\right), y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 2e+287)
   (fma (fma z_m z_m (- 0.0 t)) (* y -4.0) (* x x))
   (* z_m (* z_m (* y -4.0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 2e+287) {
		tmp = fma(fma(z_m, z_m, (0.0 - t)), (y * -4.0), (x * x));
	} else {
		tmp = z_m * (z_m * (y * -4.0));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+287)
		tmp = fma(fma(z_m, z_m, Float64(0.0 - t)), Float64(y * -4.0), Float64(x * x));
	else
		tmp = Float64(z_m * Float64(z_m * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+287], N[(N[(z$95$m * z$95$m + N[(0.0 - t), $MachinePrecision]), $MachinePrecision] * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000002e287

   1. Initial program 97.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

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

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, \color{blue}{0 - t}\right), \mathsf{neg}\left(y \cdot 4\right), x \cdot x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, \color{blue}{0 - t}\right), \mathsf{neg}\left(y \cdot 4\right), x \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), \color{blue}{y \cdot \left(\mathsf{neg}\left(4\right)\right)}, x \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), \color{blue}{y \cdot \left(\mathsf{neg}\left(4\right)\right)}, x \cdot x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot \color{blue}{-4}, x \cdot x\right) \]

      13. *-lowering-*.f64 99.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, \color{blue}{x \cdot x}\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, x \cdot x\right)} \]

   if 2.0000000000000002e287 < (*.f64 z z)

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot {z}^{2}, 0\right)} \]

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

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{y \cdot {z}^{2}}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\left({z}^{2} + 0\right)}, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \left(\color{blue}{z \cdot z} + 0\right), 0\right) \]

      6. accelerator-lowering-fma.f64 74.6

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 0\right) \]

   5. Simplified 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot \mathsf{fma}\left(z, z, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot z + 0\right)\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 84.9

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

   7. Applied egg-rr 84.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(\mathsf{fma}\left(z\_m, z\_m, 0 - t\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 2e+287)
   (fma x x (* y (* (fma z_m z_m (- 0.0 t)) -4.0)))
   (* z_m (* z_m (* y -4.0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 2e+287) {
		tmp = fma(x, x, (y * (fma(z_m, z_m, (0.0 - t)) * -4.0)));
	} else {
		tmp = z_m * (z_m * (y * -4.0));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+287)
		tmp = fma(x, x, Float64(y * Float64(fma(z_m, z_m, Float64(0.0 - t)) * -4.0)));
	else
		tmp = Float64(z_m * Float64(z_m * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+287], N[(x * x + N[(y * N[(N[(z$95$m * z$95$m + N[(0.0 - t), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000002e287

   1. Initial program 97.8%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, x, y \cdot \left(\mathsf{fma}\left(z, z, \color{blue}{0 - t}\right) \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, x, y \cdot \left(\mathsf{fma}\left(z, z, \color{blue}{0 - t}\right) \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right) \]

      13. metadata-eval 98.3

          \[\leadsto \mathsf{fma}\left(x, x, y \cdot \left(\mathsf{fma}\left(z, z, 0 - t\right) \cdot \color{blue}{-4}\right)\right) \]

   4. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot \left(\mathsf{fma}\left(z, z, 0 - t\right) \cdot -4\right)\right)} \]

   if 2.0000000000000002e287 < (*.f64 z z)

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot {z}^{2}, 0\right)} \]

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

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{y \cdot {z}^{2}}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\left({z}^{2} + 0\right)}, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \left(\color{blue}{z \cdot z} + 0\right), 0\right) \]

      6. accelerator-lowering-fma.f64 74.6

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 0\right) \]

   5. Simplified 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot \mathsf{fma}\left(z, z, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot z + 0\right)\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 84.9

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

   7. Applied egg-rr 84.9%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(\mathsf{fma}\left(z, z, 0 - t\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z\_m, z\_m, 0 - t\right), y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \frac{z\_m}{\frac{-0.25}{y}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 3.6e+151)
   (fma (fma z_m z_m (- 0.0 t)) (* y -4.0) (* x x))
   (* z_m (/ z_m (/ -0.25 y)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.6e+151) {
		tmp = fma(fma(z_m, z_m, (0.0 - t)), (y * -4.0), (x * x));
	} else {
		tmp = z_m * (z_m / (-0.25 / y));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.6e+151)
		tmp = fma(fma(z_m, z_m, Float64(0.0 - t)), Float64(y * -4.0), Float64(x * x));
	else
		tmp = Float64(z_m * Float64(z_m / Float64(-0.25 / y)));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 3.6e+151], N[(N[(z$95$m * z$95$m + N[(0.0 - t), $MachinePrecision]), $MachinePrecision] * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(z$95$m / N[(-0.25 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.6e151

   1. Initial program 94.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

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

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, \color{blue}{0 - t}\right), \mathsf{neg}\left(y \cdot 4\right), x \cdot x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, \color{blue}{0 - t}\right), \mathsf{neg}\left(y \cdot 4\right), x \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), \color{blue}{y \cdot \left(\mathsf{neg}\left(4\right)\right)}, x \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), \color{blue}{y \cdot \left(\mathsf{neg}\left(4\right)\right)}, x \cdot x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot \color{blue}{-4}, x \cdot x\right) \]

      13. *-lowering-*.f64 95.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, \color{blue}{x \cdot x}\right) \]

   4. Applied egg-rr 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, x \cdot x\right)} \]

   if 3.6e151 < z

   1. Initial program 61.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot {z}^{2}, 0\right)} \]

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

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{y \cdot {z}^{2}}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\left({z}^{2} + 0\right)}, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \left(\color{blue}{z \cdot z} + 0\right), 0\right) \]

      6. accelerator-lowering-fma.f64 67.4

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 0\right) \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot \mathsf{fma}\left(z, z, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot z + 0\right)\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 90.5

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

   7. Applied egg-rr 90.5%

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

      8. associate-/r* N/A

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

      9. *-inverses N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{z}{\frac{\color{blue}{\frac{-4}{16}}}{y}} \cdot z \]

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{z}{\frac{\frac{-4}{\color{blue}{16}}}{y}} \cdot z \]

      17. metadata-eval 90.6

          \[\leadsto \frac{z}{\frac{\color{blue}{-0.25}}{y}} \cdot z \]

   9. Applied egg-rr 90.6%

      \[\leadsto \color{blue}{\frac{z}{\frac{-0.25}{y}}} \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0 - t\right), y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z}{\frac{-0.25}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 1e+155)
   (fma y (* t 4.0) (* x x))
   (* z_m (* z_m (* y -4.0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 1e+155) {
		tmp = fma(y, (t * 4.0), (x * x));
	} else {
		tmp = z_m * (z_m * (y * -4.0));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e+155)
		tmp = fma(y, Float64(t * 4.0), Float64(x * x));
	else
		tmp = Float64(z_m * Float64(z_m * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+155], N[(y * N[(t * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000001e155

   1. Initial program 98.1%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(y, t \cdot 4, \color{blue}{{x}^{2} + 0}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, t \cdot 4, \color{blue}{x \cdot x} + 0\right) \]

      11. accelerator-lowering-fma.f64 91.0

          \[\leadsto \mathsf{fma}\left(y, t \cdot 4, \color{blue}{\mathsf{fma}\left(x, x, 0\right)}\right) \]

   5. Simplified 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot 4, \mathsf{fma}\left(x, x, 0\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 91.0

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

   7. Applied egg-rr 91.0%

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

   if 1.00000000000000001e155 < (*.f64 z z)

   1. Initial program 77.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot {z}^{2}, 0\right)} \]

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

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{y \cdot {z}^{2}}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\left({z}^{2} + 0\right)}, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \left(\color{blue}{z \cdot z} + 0\right), 0\right) \]

      6. accelerator-lowering-fma.f64 75.8

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 0\right) \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot \mathsf{fma}\left(z, z, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot z + 0\right)\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 83.2

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

   7. Applied egg-rr 83.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(t \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 5e+77) (fma x x (* 4.0 (* t y))) (* z_m (* z_m (* y -4.0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 5e+77) {
		tmp = fma(x, x, (4.0 * (t * y)));
	} else {
		tmp = z_m * (z_m * (y * -4.0));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 5e+77)
		tmp = fma(x, x, Float64(4.0 * Float64(t * y)));
	else
		tmp = Float64(z_m * Float64(z_m * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 5e+77], N[(x * x + N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.00000000000000004e77

   1. Initial program 94.5%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(y, t \cdot 4, \color{blue}{{x}^{2} + 0}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, t \cdot 4, \color{blue}{x \cdot x} + 0\right) \]

      11. accelerator-lowering-fma.f64 77.8

          \[\leadsto \mathsf{fma}\left(y, t \cdot 4, \color{blue}{\mathsf{fma}\left(x, x, 0\right)}\right) \]

   5. Simplified 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot 4, \mathsf{fma}\left(x, x, 0\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-commutative N/A

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

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

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

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(t \cdot 4\right) + 0}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(y, t \cdot 4, 0\right)}\right) \]

      6. *-lowering-*.f64 75.9

         \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, 0\right)\right) \]

   7. Applied egg-rr 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, t \cdot 4, 0\right)\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 75.9

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

   9. Applied egg-rr 75.9%

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

   if 5.00000000000000004e77 < z

   1. Initial program 70.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot {z}^{2}, 0\right)} \]

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

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{y \cdot {z}^{2}}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\left({z}^{2} + 0\right)}, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \left(\color{blue}{z \cdot z} + 0\right), 0\right) \]

      6. accelerator-lowering-fma.f64 67.1

         \[\leadsto \mathsf{fma}\left(-4, y \cdot \color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 0\right) \]

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, y \cdot \mathsf{fma}\left(z, z, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot z + 0\right)\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 83.2

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

   7. Applied egg-rr 83.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(t \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* x x) 2e+67) (* t (* y 4.0)) (* x x)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((x * x) <= 2e+67) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 2d+67) then
        tmp = t * (y * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((x * x) <= 2e+67) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if (x * x) <= 2e+67:
		tmp = t * (y * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(x * x) <= 2e+67)
		tmp = Float64(t * Float64(y * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if ((x * x) <= 2e+67)
		tmp = t * (y * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+67], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999997e67

   1. Initial program 94.0%

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

   3. Taylor expanded in t around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right) + 0} \]

      2. *-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(y \cdot t\right)} + 0 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(4 \cdot y\right) \cdot t} + 0 \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot 4\right)} \cdot t + 0 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(4 \cdot t\right)} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 4 \cdot t, 0\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, 0\right) \]

      8. *-lowering-*.f64 50.3

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot 4}, 0\right) \]

   5. Simplified 50.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot 4, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 50.3

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

   7. Applied egg-rr 50.3%

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

   if 1.99999999999999997e67 < (*.f64 x x)

   1. Initial program 85.5%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 74.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   5. Simplified 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 74.3

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

   7. Applied egg-rr 74.3%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x \cdot x \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t) :precision binary64 (* x x))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	return x * x;
}

Fortran

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

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	return x * x;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	return x * x

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	return Float64(x * x)
end

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := N[(x * x), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{{x}^{2} + 0} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{x \cdot x} + 0 \]

   3. accelerator-lowering-fma.f64 40.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

5. Simplified 40.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 40.7

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

7. Applied egg-rr 40.7%

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

Developer Target 1: 90.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(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))))