math.cube on complex, real part

Percentage Accurate: 82.2% → 99.7%

Time: 13.0s

Alternatives: 14

Speedup: 1.2×

• 44.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (-
  (* (- (* x.re x.re) (* x.im x.im)) x.re)
  (* (+ (* x.re x.im) (* x.im x.re)) x.im)))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (x_46im * x_46re)) * x_46im)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (-
  (* (- (* x.re x.re) (* x.im x.im)) x.re)
  (* (+ (* x.re x.im) (* x.im x.re)) x.im)))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (x_46im * x_46re)) * x_46im)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot -3, x.im\_m \cdot x.im\_m, x.re \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.re, \left(-x.im\_m\right) - x.im\_m, \left(x.re - x.im\_m\right) \cdot \mathsf{fma}\left(x.re, \frac{x.re}{x.im\_m}, x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 1e-12)
   (fma (* x.re -3.0) (* x.im_m x.im_m) (* x.re (* x.re x.re)))
   (*
    x.im_m
    (fma
     x.re
     (- (- x.im_m) x.im_m)
     (* (- x.re x.im_m) (fma x.re (/ x.re x.im_m) x.re))))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1e-12) {
		tmp = fma((x_46_re * -3.0), (x_46_im_m * x_46_im_m), (x_46_re * (x_46_re * x_46_re)));
	} else {
		tmp = x_46_im_m * fma(x_46_re, (-x_46_im_m - x_46_im_m), ((x_46_re - x_46_im_m) * fma(x_46_re, (x_46_re / x_46_im_m), x_46_re)));
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1e-12)
		tmp = fma(Float64(x_46_re * -3.0), Float64(x_46_im_m * x_46_im_m), Float64(x_46_re * Float64(x_46_re * x_46_re)));
	else
		tmp = Float64(x_46_im_m * fma(x_46_re, Float64(Float64(-x_46_im_m) - x_46_im_m), Float64(Float64(x_46_re - x_46_im_m) * fma(x_46_re, Float64(x_46_re / x_46_im_m), x_46_re))));
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 1e-12], N[(N[(x$46$re * -3.0), $MachinePrecision] * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re * N[((-x$46$im$95$m) - x$46$im$95$m), $MachinePrecision] + N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$re * N[(x$46$re / x$46$im$95$m), $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 9.9999999999999998e-13

   1. Initial program 99.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{x.re \cdot \left(\left(-1 \cdot {x.im}^{2} + {x.re}^{2}\right) - 2 \cdot {x.im}^{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x.re \cdot \left(\left(-1 \cdot {x.im}^{2} + {x.re}^{2}\right) - 2 \cdot {x.im}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto x.re \cdot \left(\color{blue}{\left({x.re}^{2} + -1 \cdot {x.im}^{2}\right)} - 2 \cdot {x.im}^{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto x.re \cdot \color{blue}{\left({x.re}^{2} + \left(-1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto x.re \cdot \left(\color{blue}{x.re \cdot x.re} + \left(-1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x.re \cdot \color{blue}{\mathsf{fma}\left(x.re, x.re, -1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)} \]

      6. distribute-rgt-out-- N/A

         \[\leadsto x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{{x.im}^{2} \cdot \left(-1 - 2\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{{x.im}^{2} \cdot \left(-1 - 2\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(-1 - 2\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(-1 - 2\right)\right) \]

      10. metadata-eval 99.8

          \[\leadsto x.re \cdot \mathsf{fma}\left(x.re, x.re, \left(x.im \cdot x.im\right) \cdot \color{blue}{-3}\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{x.re \cdot \mathsf{fma}\left(x.re, x.re, \left(x.im \cdot x.im\right) \cdot -3\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x.re \cdot \left(x.re \cdot x.re + \color{blue}{\left(x.im \cdot x.im\right)} \cdot -3\right) \]

      2. lift-*.f64 N/A

         \[\leadsto x.re \cdot \left(x.re \cdot x.re + \color{blue}{\left(x.im \cdot x.im\right) \cdot -3}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x.re \cdot \left(\color{blue}{x.re \cdot x.re} + \left(x.im \cdot x.im\right) \cdot -3\right) \]

      4. +-commutative N/A

         \[\leadsto x.re \cdot \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot -3 + x.re \cdot x.re\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right) + x.re \cdot \left(x.re \cdot x.re\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto x.re \cdot \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot -3\right)} + x.re \cdot \left(x.re \cdot x.re\right) \]

      7. *-commutative N/A

         \[\leadsto x.re \cdot \color{blue}{\left(-3 \cdot \left(x.im \cdot x.im\right)\right)} + x.re \cdot \left(x.re \cdot x.re\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)} + x.re \cdot \left(x.re \cdot x.re\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right) + \color{blue}{x.re \cdot \left(x.re \cdot x.re\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot -3, x.im \cdot x.im, x.re \cdot \left(x.re \cdot x.re\right)\right)} \]

      11. lower-*.f64 99.8

          \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot -3}, x.im \cdot x.im, x.re \cdot \left(x.re \cdot x.re\right)\right) \]

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot -3, x.im \cdot x.im, x.re \cdot \left(x.re \cdot x.re\right)\right)} \]

   if 9.9999999999999998e-13 < x.im

   1. Initial program 66.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing

   4. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)} \]

   5. Taylor expanded in x.im around inf

      \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.im \cdot \left(x.re + \frac{{x.re}^{2}}{x.im}\right)\right)} \cdot \left(x.re - x.im\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.im \cdot \left(x.re + \frac{{x.re}^{2}}{x.im}\right)\right)} \cdot \left(x.re - x.im\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(x.im \cdot \color{blue}{\left(\frac{{x.re}^{2}}{x.im} + x.re\right)}\right) \cdot \left(x.re - x.im\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(x.im \cdot \left(\frac{\color{blue}{x.re \cdot x.re}}{x.im} + x.re\right)\right) \cdot \left(x.re - x.im\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(x.im \cdot \left(\color{blue}{x.re \cdot \frac{x.re}{x.im}} + x.re\right)\right) \cdot \left(x.re - x.im\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(x.im \cdot \color{blue}{\mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.re\right)}\right) \cdot \left(x.re - x.im\right)\right) \]

      6. lower-/.f64 86.0

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.im \cdot \mathsf{fma}\left(x.re, \color{blue}{\frac{x.re}{x.im}}, x.re\right)\right) \cdot \left(x.re - x.im\right)\right) \]

   7. Simplified 86.0%

      \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \color{blue}{\left(x.im \cdot \mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.re\right)\right)} \cdot \left(x.re - x.im\right)\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(x.im \cdot \left(x.re \cdot \color{blue}{\frac{x.re}{x.im}} + x.re\right)\right) \cdot \left(x.re - x.im\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(\left(x.re \cdot \frac{x.re}{x.im}\right) \cdot x.im + x.re \cdot x.im\right)} \cdot \left(x.re - x.im\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(\left(x.re \cdot \frac{x.re}{x.im}\right) \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot \left(x.re - x.im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(\color{blue}{\left(\frac{x.re}{x.im} \cdot x.re\right)} \cdot x.im + x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(\color{blue}{\frac{x.re}{x.im} \cdot \left(x.re \cdot x.im\right)} + x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(\frac{x.re}{x.im} \cdot \color{blue}{\left(x.re \cdot x.im\right)} + x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\right) \]

      7. lower-fma.f64 81.1

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \color{blue}{\mathsf{fma}\left(\frac{x.re}{x.im}, x.re \cdot x.im, x.re \cdot x.im\right)} \cdot \left(x.re - x.im\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(\frac{x.re}{x.im}, \color{blue}{x.re \cdot x.im}, x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(\frac{x.re}{x.im}, \color{blue}{x.im \cdot x.re}, x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\right) \]

      10. lower-*.f64 81.1

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \mathsf{fma}\left(\frac{x.re}{x.im}, \color{blue}{x.im \cdot x.re}, x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(\frac{x.re}{x.im}, x.im \cdot x.re, \color{blue}{x.re \cdot x.im}\right) \cdot \left(x.re - x.im\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(\frac{x.re}{x.im}, x.im \cdot x.re, \color{blue}{x.im \cdot x.re}\right) \cdot \left(x.re - x.im\right)\right) \]

      13. lower-*.f64 81.1

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \mathsf{fma}\left(\frac{x.re}{x.im}, x.im \cdot x.re, \color{blue}{x.im \cdot x.re}\right) \cdot \left(x.re - x.im\right)\right) \]

   9. Applied egg-rr 81.1%

      \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \color{blue}{\mathsf{fma}\left(\frac{x.re}{x.im}, x.im \cdot x.re, x.im \cdot x.re\right)} \cdot \left(x.re - x.im\right)\right) \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \left(x.re \cdot \color{blue}{\left(x.im + x.im\right)}\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(\frac{x.re}{x.im} \cdot \left(x.im \cdot x.re\right) + x.im \cdot x.re\right) \cdot \left(x.re - x.im\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(\frac{x.re}{x.im} \cdot \left(x.im \cdot x.re\right) + x.im \cdot x.re\right) \cdot \left(x.re - x.im\right) \]

       3. lift-neg.f64 N/A

          \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)} + \left(\frac{x.re}{x.im} \cdot \left(x.im \cdot x.re\right) + x.im \cdot x.re\right) \cdot \left(x.re - x.im\right) \]

       4. lift-/.f64 N/A

          \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(\color{blue}{\frac{x.re}{x.im}} \cdot \left(x.im \cdot x.re\right) + x.im \cdot x.re\right) \cdot \left(x.re - x.im\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(\frac{x.re}{x.im} \cdot \color{blue}{\left(x.im \cdot x.re\right)} + x.im \cdot x.re\right) \cdot \left(x.re - x.im\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(\frac{x.re}{x.im} \cdot \left(x.im \cdot x.re\right) + \color{blue}{x.im \cdot x.re}\right) \cdot \left(x.re - x.im\right) \]

       7. lift-fma.f64 N/A

          \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \color{blue}{\mathsf{fma}\left(\frac{x.re}{x.im}, x.im \cdot x.re, x.im \cdot x.re\right)} \cdot \left(x.re - x.im\right) \]

       8. lift--.f64 N/A

          \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \mathsf{fma}\left(\frac{x.re}{x.im}, x.im \cdot x.re, x.im \cdot x.re\right) \cdot \color{blue}{\left(x.re - x.im\right)} \]

       9. *-commutative N/A

          \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \color{blue}{\left(x.re - x.im\right) \cdot \mathsf{fma}\left(\frac{x.re}{x.im}, x.im \cdot x.re, x.im \cdot x.re\right)} \]

       10. lift-fma.f64 N/A

           \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re - x.im\right) \cdot \color{blue}{\left(\frac{x.re}{x.im} \cdot \left(x.im \cdot x.re\right) + x.im \cdot x.re\right)} \]

       11. distribute-lft1-in N/A

           \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re - x.im\right) \cdot \color{blue}{\left(\left(\frac{x.re}{x.im} + 1\right) \cdot \left(x.im \cdot x.re\right)\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re - x.im\right) \cdot \left(\left(\frac{x.re}{x.im} + 1\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re - x.im\right) \cdot \left(\left(\frac{x.re}{x.im} + 1\right) \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right) \]

       14. associate-*r* N/A

           \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re - x.im\right) \cdot \color{blue}{\left(\left(\left(\frac{x.re}{x.im} + 1\right) \cdot x.re\right) \cdot x.im\right)} \]

       15. distribute-lft1-in N/A

           \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re - x.im\right) \cdot \left(\color{blue}{\left(\frac{x.re}{x.im} \cdot x.re + x.re\right)} \cdot x.im\right) \]

       16. *-commutative N/A

           \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re - x.im\right) \cdot \left(\left(\color{blue}{x.re \cdot \frac{x.re}{x.im}} + x.re\right) \cdot x.im\right) \]

       17. lift-fma.f64 N/A

           \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re - x.im\right) \cdot \left(\color{blue}{\mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.re\right)} \cdot x.im\right) \]

       18. associate-*l* N/A

           \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \color{blue}{\left(\left(x.re - x.im\right) \cdot \mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.re\right)\right) \cdot x.im} \]

   11. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(x.re, -\left(x.im + x.im\right), \left(x.re - x.im\right) \cdot \mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot -3, x.im \cdot x.im, x.re \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.re, \left(-x.im\right) - x.im, \left(x.re - x.im\right) \cdot \mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+81}:\\ \;\;\;\;x.im\_m \cdot \left(-3 \cdot \left(x.im\_m \cdot x.re\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-179}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, -3 \cdot \left(x.im\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(x.re \cdot \left(x.re - x.im\_m\right)\right) - \left(x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (if (<= t_0 -5e+81)
     (* x.im_m (* -3.0 (* x.im_m x.re)))
     (if (<= t_0 2e-179)
       (* x.re (fma x.re x.re (* -3.0 (* x.im_m x.im_m))))
       (- (* (+ x.im_m x.re) (* x.re (- x.re x.im_m))) (+ x.im_m x.im_m))))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_0 <= -5e+81) {
		tmp = x_46_im_m * (-3.0 * (x_46_im_m * x_46_re));
	} else if (t_0 <= 2e-179) {
		tmp = x_46_re * fma(x_46_re, x_46_re, (-3.0 * (x_46_im_m * x_46_im_m)));
	} else {
		tmp = ((x_46_im_m + x_46_re) * (x_46_re * (x_46_re - x_46_im_m))) - (x_46_im_m + x_46_im_m);
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_0 <= -5e+81)
		tmp = Float64(x_46_im_m * Float64(-3.0 * Float64(x_46_im_m * x_46_re)));
	elseif (t_0 <= 2e-179)
		tmp = Float64(x_46_re * fma(x_46_re, x_46_re, Float64(-3.0 * Float64(x_46_im_m * x_46_im_m))));
	else
		tmp = Float64(Float64(Float64(x_46_im_m + x_46_re) * Float64(x_46_re * Float64(x_46_re - x_46_im_m))) - Float64(x_46_im_m + x_46_im_m));
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+81], N[(x$46$im$95$m * N[(-3.0 * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-179], N[(x$46$re * N[(x$46$re * x$46$re + N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.9999999999999998e81

   1. Initial program 89.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)} \]

   5. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{x.re \cdot \left(-2 \cdot {x.im}^{2} + -1 \cdot {x.im}^{2}\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-out N/A

         \[\leadsto x.re \cdot \color{blue}{\left({x.im}^{2} \cdot \left(-2 + -1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x.re \cdot \left({x.im}^{2} \cdot \color{blue}{-3}\right) \]

      3. *-commutative N/A

         \[\leadsto x.re \cdot \color{blue}{\left(-3 \cdot {x.im}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(-3 \cdot {x.im}^{2}\right) \cdot x.re} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{-3 \cdot \left({x.im}^{2} \cdot x.re\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{-3 \cdot \left({x.im}^{2} \cdot x.re\right)} \]

      7. *-commutative N/A

         \[\leadsto -3 \cdot \color{blue}{\left(x.re \cdot {x.im}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto -3 \cdot \color{blue}{\left(x.re \cdot {x.im}^{2}\right)} \]

      9. unpow2 N/A

         \[\leadsto -3 \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

      10. lower-*.f64 37.2

          \[\leadsto -3 \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

   7. Simplified 37.2%

      \[\leadsto \color{blue}{-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -3 \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto -3 \cdot \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot x.re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto -3 \cdot \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot x.re\right) \]

      4. associate-*l* N/A

         \[\leadsto -3 \cdot \color{blue}{\left(x.im \cdot \left(x.im \cdot x.re\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto -3 \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto -3 \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(-3 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(x.im \cdot -3\right)} \cdot \left(x.re \cdot x.im\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x.im \cdot -3\right)} \cdot \left(x.re \cdot x.im\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(x.im \cdot -3\right) \cdot \color{blue}{\left(x.re \cdot x.im\right)} \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x.im \cdot -3\right) \cdot x.re\right) \cdot x.im} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(x.im \cdot -3\right) \cdot x.re\right) \cdot x.im} \]

      13. lower-*.f64 48.1

          \[\leadsto \color{blue}{\left(\left(x.im \cdot -3\right) \cdot x.re\right)} \cdot x.im \]

   9. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{\left(\left(x.im \cdot -3\right) \cdot x.re\right) \cdot x.im} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(x.im \cdot -3\right)} \cdot x.re\right) \cdot x.im \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(x.re \cdot \left(x.im \cdot -3\right)\right)} \cdot x.im \]

       3. lift-*.f64 N/A

          \[\leadsto \left(x.re \cdot \color{blue}{\left(x.im \cdot -3\right)}\right) \cdot x.im \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot -3\right)} \cdot x.im \]

       5. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x.im \cdot x.re\right)} \cdot -3\right) \cdot x.im \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(x.im \cdot x.re\right)} \cdot -3\right) \cdot x.im \]

       7. lower-*.f64 48.1

          \[\leadsto \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot -3\right)} \cdot x.im \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(x.im \cdot x.re\right)} \cdot -3\right) \cdot x.im \]

       9. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x.re \cdot x.im\right)} \cdot -3\right) \cdot x.im \]

       10. lift-*.f64 48.1

           \[\leadsto \left(\color{blue}{\left(x.re \cdot x.im\right)} \cdot -3\right) \cdot x.im \]

   11. Applied egg-rr 48.1%

       \[\leadsto \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot -3\right)} \cdot x.im \]

   if -4.9999999999999998e81 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 2e-179

   1. Initial program 99.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{x.re \cdot \left(\left(-1 \cdot {x.im}^{2} + {x.re}^{2}\right) - 2 \cdot {x.im}^{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x.re \cdot \left(\left(-1 \cdot {x.im}^{2} + {x.re}^{2}\right) - 2 \cdot {x.im}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto x.re \cdot \left(\color{blue}{\left({x.re}^{2} + -1 \cdot {x.im}^{2}\right)} - 2 \cdot {x.im}^{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto x.re \cdot \color{blue}{\left({x.re}^{2} + \left(-1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto x.re \cdot \left(\color{blue}{x.re \cdot x.re} + \left(-1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x.re \cdot \color{blue}{\mathsf{fma}\left(x.re, x.re, -1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)} \]

      6. distribute-rgt-out-- N/A

         \[\leadsto x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{{x.im}^{2} \cdot \left(-1 - 2\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{{x.im}^{2} \cdot \left(-1 - 2\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(-1 - 2\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(-1 - 2\right)\right) \]

      10. metadata-eval 99.7

          \[\leadsto x.re \cdot \mathsf{fma}\left(x.re, x.re, \left(x.im \cdot x.im\right) \cdot \color{blue}{-3}\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{x.re \cdot \mathsf{fma}\left(x.re, x.re, \left(x.im \cdot x.im\right) \cdot -3\right)} \]

   if 2e-179 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 67.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing

   4. Applied egg-rr 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(x.re \cdot \color{blue}{\left(x.im + x.im\right)}\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)} + \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot \color{blue}{\left(x.re + x.im\right)}\right) \cdot \left(x.re - x.im\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right)} \cdot \left(x.re - x.im\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \color{blue}{\left(x.re - x.im\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right) \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      10. lift-neg.f64 N/A

          \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)} \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

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

          \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      13. lift--.f64 81.2

          \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   6. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - \left(x.im + x.im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -5 \cdot 10^{+81}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 2 \cdot 10^{-179}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - \left(x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 87.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024218 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))