math.cube on complex, imaginary part

Percentage Accurate: 82.3% → 96.5%

Time: 12.2s

Alternatives: 8

Speedup: 1.2×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.3% accurate, 1.0× speedup

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.18 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.re\_m \cdot x.re\_m, 3, 0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im \cdot \left(x.re\_m \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 1.18e+154)
   (* x.im (fma (* x.re_m x.re_m) 3.0 (- 0.0 (* x.im x.im))))
   (* x.re_m (* x.im (* x.re_m 3.0)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.18e+154) {
		tmp = x_46_im * fma((x_46_re_m * x_46_re_m), 3.0, (0.0 - (x_46_im * x_46_im)));
	} else {
		tmp = x_46_re_m * (x_46_im * (x_46_re_m * 3.0));
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1.18e+154)
		tmp = Float64(x_46_im * fma(Float64(x_46_re_m * x_46_re_m), 3.0, Float64(0.0 - Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im * Float64(x_46_re_m * 3.0)));
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 1.18e+154], N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 3.0 + N[(0.0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.18000000000000004e154

   1. Initial program 86.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 92.5%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 92.5%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(x.re \cdot \left(x.re \cdot 3\right) + \color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(x.re \cdot x.re\right) \cdot 3 + \left(\mathsf{neg}\left(\color{blue}{x.im \cdot x.im}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{3}, \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3, \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3, \left(0 - x.im \cdot x.im\right)\right)\right) \]

      6. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3, \left(\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right) - x.im \cdot x.im\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3, \left(\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.im \cdot x.re\right)\right) - x.im \cdot x.im\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3, \left(\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) - x.im \cdot x.im\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.im \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right), \left(x.im \cdot x.im\right)\right)\right)\right) \]

      12. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3, \mathsf{\_.f64}\left(0, \left(x.im \cdot x.im\right)\right)\right)\right) \]

      13. *-lowering-*.f64 93.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right) \]

   6. Applied egg-rr 93.0%

      \[\leadsto x.im \cdot \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, 3, 0 - x.im \cdot x.im\right)} \]

   if 1.18000000000000004e154 < x.re

   1. Initial program 54.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 54.9%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 54.9%

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

   5. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

         \[\leadsto \left(3 \cdot {x.re}^{2}\right) \cdot \left(x.im \cdot \color{blue}{1}\right) \]

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

         \[\leadsto \frac{3 \cdot {x.re}^{2}}{{x.im}^{2}} \cdot \color{blue}{{x.im}^{3}} \]

      10. associate-*r/ N/A

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

      11. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right)}\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\frac{{x.re}^{2} \cdot {x.im}^{3}}{\color{blue}{{x.im}^{2}}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left({x.re}^{2} \cdot \color{blue}{\frac{{x.im}^{3}}{{x.im}^{2}}}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{{x.im}^{3}}}{{x.im}^{2}}\right)\right) \]

      16. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot \left(x.im \cdot x.im\right)}{{\color{blue}{x.im}}^{2}}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot {x.im}^{2}}{{x.im}^{2}}\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot \color{blue}{\frac{{x.im}^{2}}{{x.im}^{2}}}\right)\right)\right) \]

      19. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 1\right)\right)\right) \]

      20. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      24. *-lowering-*.f64 83.1%

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 83.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x.re \cdot 3\right) \cdot x.im\right), \color{blue}{x.re}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot 3\right), x.im\right), x.re\right) \]

      6. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, 3\right), x.im\right), x.re\right) \]

   9. Applied egg-rr 83.3%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.18 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.re \cdot x.re, 3, 0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im \cdot \left(x.re\_m \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 7.6e+153)
   (* x.im (- (* x.re_m (* x.re_m 3.0)) (* x.im x.im)))
   (* x.re_m (* x.im (* x.re_m 3.0)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 7.6e+153) {
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_im * (x_46_re_m * 3.0));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 7.6d+153) then
        tmp = x_46im * ((x_46re_m * (x_46re_m * 3.0d0)) - (x_46im * x_46im))
    else
        tmp = x_46re_m * (x_46im * (x_46re_m * 3.0d0))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 7.6e+153) {
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_im * (x_46_re_m * 3.0));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 7.6e+153:
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im))
	else:
		tmp = x_46_re_m * (x_46_im * (x_46_re_m * 3.0))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 7.6e+153)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re_m * Float64(x_46_re_m * 3.0)) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im * Float64(x_46_re_m * 3.0)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 7.6e+153)
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	else
		tmp = x_46_re_m * (x_46_im * (x_46_re_m * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 7.6e+153], N[(x$46$im * N[(N[(x$46$re$95$m * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.59999999999999933e153

   1. Initial program 86.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 92.9%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 92.9%

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

   if 7.59999999999999933e153 < x.re

   1. Initial program 53.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 53.1%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 53.1%

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

   5. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

         \[\leadsto \left(3 \cdot {x.re}^{2}\right) \cdot \left(x.im \cdot \color{blue}{1}\right) \]

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

         \[\leadsto \frac{3 \cdot {x.re}^{2}}{{x.im}^{2}} \cdot \color{blue}{{x.im}^{3}} \]

      10. associate-*r/ N/A

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

      11. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right)}\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\frac{{x.re}^{2} \cdot {x.im}^{3}}{\color{blue}{{x.im}^{2}}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left({x.re}^{2} \cdot \color{blue}{\frac{{x.im}^{3}}{{x.im}^{2}}}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{{x.im}^{3}}}{{x.im}^{2}}\right)\right) \]

      16. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot \left(x.im \cdot x.im\right)}{{\color{blue}{x.im}}^{2}}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot {x.im}^{2}}{{x.im}^{2}}\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot \color{blue}{\frac{{x.im}^{2}}{{x.im}^{2}}}\right)\right)\right) \]

      19. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 1\right)\right)\right) \]

      20. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      24. *-lowering-*.f64 80.4%

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 80.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x.re \cdot 3\right) \cdot x.im\right), \color{blue}{x.re}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot 3\right), x.im\right), x.re\right) \]

      6. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, 3\right), x.im\right), x.re\right) \]

   9. Applied egg-rr 80.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 6.5 \cdot 10^{-14}:\\ \;\;\;\;x.re\_m \cdot \left(x.im \cdot \left(x.re\_m \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 6.5e-14)
   (* x.re_m (* x.im (* x.re_m 3.0)))
   (- 0.0 (* x.im (* x.im x.im)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 6.5e-14) {
		tmp = x_46_re_m * (x_46_im * (x_46_re_m * 3.0));
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 6.5d-14) then
        tmp = x_46re_m * (x_46im * (x_46re_m * 3.0d0))
    else
        tmp = 0.0d0 - (x_46im * (x_46im * x_46im))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 6.5e-14) {
		tmp = x_46_re_m * (x_46_im * (x_46_re_m * 3.0));
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 6.5e-14:
		tmp = x_46_re_m * (x_46_im * (x_46_re_m * 3.0))
	else:
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 6.5e-14)
		tmp = Float64(x_46_re_m * Float64(x_46_im * Float64(x_46_re_m * 3.0)));
	else
		tmp = Float64(0.0 - Float64(x_46_im * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 6.5e-14)
		tmp = x_46_re_m * (x_46_im * (x_46_re_m * 3.0));
	else
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 6.5e-14], N[(x$46$re$95$m * N[(x$46$im * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 6.5000000000000001e-14

   1. Initial program 83.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 87.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 87.0%

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

   5. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

         \[\leadsto \left(3 \cdot {x.re}^{2}\right) \cdot \left(x.im \cdot \color{blue}{1}\right) \]

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

         \[\leadsto \frac{3 \cdot {x.re}^{2}}{{x.im}^{2}} \cdot \color{blue}{{x.im}^{3}} \]

      10. associate-*r/ N/A

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

      11. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right)}\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\frac{{x.re}^{2} \cdot {x.im}^{3}}{\color{blue}{{x.im}^{2}}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left({x.re}^{2} \cdot \color{blue}{\frac{{x.im}^{3}}{{x.im}^{2}}}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{{x.im}^{3}}}{{x.im}^{2}}\right)\right) \]

      16. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot \left(x.im \cdot x.im\right)}{{\color{blue}{x.im}}^{2}}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot {x.im}^{2}}{{x.im}^{2}}\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot \color{blue}{\frac{{x.im}^{2}}{{x.im}^{2}}}\right)\right)\right) \]

      19. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 1\right)\right)\right) \]

      20. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      24. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 68.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x.re \cdot 3\right) \cdot x.im\right), \color{blue}{x.re}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot 3\right), x.im\right), x.re\right) \]

      6. *-lowering-*.f64 68.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, 3\right), x.im\right), x.re\right) \]

   9. Applied egg-rr 68.6%

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

   if 6.5000000000000001e-14 < x.im

   1. Initial program 79.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 91.0%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 79.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 79.4%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.im \cdot x.im\right)\right)\right) \]

      4. *-lowering-*.f64 79.4%

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

   9. Applied egg-rr 79.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 6.5 \cdot 10^{-14}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.02 \cdot 10^{-13}:\\ \;\;\;\;\left(x.re\_m \cdot 3\right) \cdot \left(x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 1.02e-13)
   (* (* x.re_m 3.0) (* x.re_m x.im))
   (- 0.0 (* x.im (* x.im x.im)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.02e-13) {
		tmp = (x_46_re_m * 3.0) * (x_46_re_m * x_46_im);
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 1.02d-13) then
        tmp = (x_46re_m * 3.0d0) * (x_46re_m * x_46im)
    else
        tmp = 0.0d0 - (x_46im * (x_46im * x_46im))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.02e-13) {
		tmp = (x_46_re_m * 3.0) * (x_46_re_m * x_46_im);
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 1.02e-13:
		tmp = (x_46_re_m * 3.0) * (x_46_re_m * x_46_im)
	else:
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.02e-13)
		tmp = Float64(Float64(x_46_re_m * 3.0) * Float64(x_46_re_m * x_46_im));
	else
		tmp = Float64(0.0 - Float64(x_46_im * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.02e-13)
		tmp = (x_46_re_m * 3.0) * (x_46_re_m * x_46_im);
	else
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 1.02e-13], N[(N[(x$46$re$95$m * 3.0), $MachinePrecision] * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.0199999999999999e-13

   1. Initial program 83.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 87.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 87.0%

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

   5. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

         \[\leadsto \left(3 \cdot {x.re}^{2}\right) \cdot \left(x.im \cdot \color{blue}{1}\right) \]

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

         \[\leadsto \frac{3 \cdot {x.re}^{2}}{{x.im}^{2}} \cdot \color{blue}{{x.im}^{3}} \]

      10. associate-*r/ N/A

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

      11. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right)}\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\frac{{x.re}^{2} \cdot {x.im}^{3}}{\color{blue}{{x.im}^{2}}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left({x.re}^{2} \cdot \color{blue}{\frac{{x.im}^{3}}{{x.im}^{2}}}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{{x.im}^{3}}}{{x.im}^{2}}\right)\right) \]

      16. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot \left(x.im \cdot x.im\right)}{{\color{blue}{x.im}}^{2}}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot {x.im}^{2}}{{x.im}^{2}}\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot \color{blue}{\frac{{x.im}^{2}}{{x.im}^{2}}}\right)\right)\right) \]

      19. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 1\right)\right)\right) \]

      20. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      24. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 68.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x.re \cdot 3\right), \color{blue}{\left(x.im \cdot x.re\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, 3\right), \left(\color{blue}{x.im} \cdot x.re\right)\right) \]

      5. *-lowering-*.f64 68.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   9. Applied egg-rr 68.6%

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

   if 1.0199999999999999e-13 < x.im

   1. Initial program 79.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 91.0%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 79.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 79.4%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.im \cdot x.im\right)\right)\right) \]

      4. *-lowering-*.f64 79.4%

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

   9. Applied egg-rr 79.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.02 \cdot 10^{-13}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;x.re\_m \cdot \left(3 \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 1.95e-13)
   (* x.re_m (* 3.0 (* x.re_m x.im)))
   (- 0.0 (* x.im (* x.im x.im)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.95e-13) {
		tmp = x_46_re_m * (3.0 * (x_46_re_m * x_46_im));
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 1.95d-13) then
        tmp = x_46re_m * (3.0d0 * (x_46re_m * x_46im))
    else
        tmp = 0.0d0 - (x_46im * (x_46im * x_46im))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.95e-13) {
		tmp = x_46_re_m * (3.0 * (x_46_re_m * x_46_im));
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 1.95e-13:
		tmp = x_46_re_m * (3.0 * (x_46_re_m * x_46_im))
	else:
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.95e-13)
		tmp = Float64(x_46_re_m * Float64(3.0 * Float64(x_46_re_m * x_46_im)));
	else
		tmp = Float64(0.0 - Float64(x_46_im * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.95e-13)
		tmp = x_46_re_m * (3.0 * (x_46_re_m * x_46_im));
	else
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 1.95e-13], N[(x$46$re$95$m * N[(3.0 * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.95000000000000002e-13

   1. Initial program 83.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 87.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 87.0%

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

   5. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

         \[\leadsto \left(3 \cdot {x.re}^{2}\right) \cdot \left(x.im \cdot \color{blue}{1}\right) \]

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

         \[\leadsto \frac{3 \cdot {x.re}^{2}}{{x.im}^{2}} \cdot \color{blue}{{x.im}^{3}} \]

      10. associate-*r/ N/A

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

      11. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right)}\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\frac{{x.re}^{2} \cdot {x.im}^{3}}{\color{blue}{{x.im}^{2}}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left({x.re}^{2} \cdot \color{blue}{\frac{{x.im}^{3}}{{x.im}^{2}}}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{{x.im}^{3}}}{{x.im}^{2}}\right)\right) \]

      16. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot \left(x.im \cdot x.im\right)}{{\color{blue}{x.im}}^{2}}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot {x.im}^{2}}{{x.im}^{2}}\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot \color{blue}{\frac{{x.im}^{2}}{{x.im}^{2}}}\right)\right)\right) \]

      19. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 1\right)\right)\right) \]

      20. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      24. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 68.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot \left(x.im \cdot x.re\right)\right), \color{blue}{x.re}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \left(x.im \cdot x.re\right)\right), x.re\right) \]

      5. *-lowering-*.f64 68.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right) \]

   9. Applied egg-rr 68.5%

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

   if 1.95000000000000002e-13 < x.im

   1. Initial program 79.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 91.0%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 79.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 79.4%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.im \cdot x.im\right)\right)\right) \]

      4. *-lowering-*.f64 79.4%

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

   9. Applied egg-rr 79.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 3.1 \cdot 10^{-13}:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 3.1e-13)
   (* 3.0 (* x.re_m (* x.re_m x.im)))
   (- 0.0 (* x.im (* x.im x.im)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.1e-13) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im));
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 3.1d-13) then
        tmp = 3.0d0 * (x_46re_m * (x_46re_m * x_46im))
    else
        tmp = 0.0d0 - (x_46im * (x_46im * x_46im))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.1e-13) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im));
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 3.1e-13:
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im))
	else:
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 3.1e-13)
		tmp = Float64(3.0 * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im)));
	else
		tmp = Float64(0.0 - Float64(x_46_im * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 3.1e-13)
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im));
	else
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 3.1e-13], N[(3.0 * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.0999999999999999e-13

   1. Initial program 83.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 87.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 87.0%

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

   5. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

         \[\leadsto \left(3 \cdot {x.re}^{2}\right) \cdot \left(x.im \cdot \color{blue}{1}\right) \]

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

         \[\leadsto \frac{3 \cdot {x.re}^{2}}{{x.im}^{2}} \cdot \color{blue}{{x.im}^{3}} \]

      10. associate-*r/ N/A

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

      11. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right)}\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\frac{{x.re}^{2} \cdot {x.im}^{3}}{\color{blue}{{x.im}^{2}}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left({x.re}^{2} \cdot \color{blue}{\frac{{x.im}^{3}}{{x.im}^{2}}}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{{x.im}^{3}}}{{x.im}^{2}}\right)\right) \]

      16. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot \left(x.im \cdot x.im\right)}{{\color{blue}{x.im}}^{2}}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \frac{x.im \cdot {x.im}^{2}}{{x.im}^{2}}\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot \color{blue}{\frac{{x.im}^{2}}{{x.im}^{2}}}\right)\right)\right) \]

      19. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 1\right)\right)\right) \]

      20. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(3, \left(x.re \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      24. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 68.5%

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

   if 3.0999999999999999e-13 < x.im

   1. Initial program 79.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 91.0%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 79.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 79.4%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.im \cdot x.im\right)\right)\right) \]

      4. *-lowering-*.f64 79.4%

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

   9. Applied egg-rr 79.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.1 \cdot 10^{-13}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{if}\;x.re\_m \leq 1.16 \cdot 10^{+205}:\\ \;\;\;\;0 - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* x.im (* x.im x.im))))
   (if (<= x.re_m 1.16e+205) (- 0.0 t_0) t_0)))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = x_46_im * (x_46_im * x_46_im);
	double tmp;
	if (x_46_re_m <= 1.16e+205) {
		tmp = 0.0 - t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im * (x_46im * x_46im)
    if (x_46re_m <= 1.16d+205) then
        tmp = 0.0d0 - t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = x_46_im * (x_46_im * x_46_im);
	double tmp;
	if (x_46_re_m <= 1.16e+205) {
		tmp = 0.0 - t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = x_46_im * (x_46_im * x_46_im)
	tmp = 0
	if x_46_re_m <= 1.16e+205:
		tmp = 0.0 - t_0
	else:
		tmp = t_0
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(x_46_im * Float64(x_46_im * x_46_im))
	tmp = 0.0
	if (x_46_re_m <= 1.16e+205)
		tmp = Float64(0.0 - t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = x_46_im * (x_46_im * x_46_im);
	tmp = 0.0;
	if (x_46_re_m <= 1.16e+205)
		tmp = 0.0 - t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re$95$m, 1.16e+205], N[(0.0 - t$95$0), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.16000000000000001e205

   1. Initial program 85.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 91.5%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 91.5%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 62.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 62.0%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.im \cdot x.im\right)\right)\right) \]

      4. *-lowering-*.f64 62.0%

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

   9. Applied egg-rr 62.0%

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

   if 1.16000000000000001e205 < x.re

   1. Initial program 51.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 51.5%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 51.5%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 19.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 19.1%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   8. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \frac{0 \cdot 0 - \left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)}{\color{blue}{0 + x.im \cdot \left(x.im \cdot x.im\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{0 - \left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{0 - \left(\left(x.im \cdot x.im\right) \cdot x.im\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{0 - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

      5. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)\right)}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(x.im \cdot x.im\right) \cdot \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

      7. cube-unmult N/A

         \[\leadsto \frac{\mathsf{neg}\left({\left(x.im \cdot x.im\right)}^{3}\right)}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

      8. cube-neg N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)}^{3}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

      9. sub0-neg N/A

         \[\leadsto \frac{{\left(0 - x.im \cdot x.im\right)}^{3}}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

      10. sqr-pow N/A

          \[\leadsto \frac{{\left(0 - x.im \cdot x.im\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - x.im \cdot x.im\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

      11. pow-prod-down N/A

          \[\leadsto \frac{{\left(\left(0 - x.im \cdot x.im\right) \cdot \left(0 - x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

      12. sub0-neg N/A

          \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) \cdot \left(0 - x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)}}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

      13. sub0-neg N/A

          \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

      14. sqr-neg N/A

          \[\leadsto \frac{{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)}}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

      15. unpow-prod-down N/A

          \[\leadsto \frac{{\left(x.im \cdot x.im\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(x.im \cdot x.im\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

      16. sqr-pow N/A

          \[\leadsto \frac{{\left(x.im \cdot x.im\right)}^{3}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

      17. unpow-prod-down N/A

          \[\leadsto \frac{{x.im}^{3} \cdot {x.im}^{3}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

      18. pow-sqr N/A

          \[\leadsto \frac{{x.im}^{\left(2 \cdot 3\right)}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

      19. +-lft-identity N/A

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

      20. cube-unmult N/A

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

   9. Applied egg-rr 15.2%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot x.im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.16 \cdot 10^{+205}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 20.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im \cdot \left(x.im \cdot x.im\right) \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im) :precision binary64 (* x.im (* x.im x.im)))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	return x_46_im * (x_46_im * x_46_im);
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46im * (x_46im * x_46im)
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	return x_46_im * (x_46_im * x_46_im);
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	return x_46_im * (x_46_im * x_46_im)

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	return Float64(x_46_im * Float64(x_46_im * x_46_im))
end

MATLAB

x.re_m = abs(x_46_re);
function tmp = code(x_46_re_m, x_46_im)
	tmp = x_46_im * (x_46_im * x_46_im);
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

   8. associate-+r- N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

   10. distribute-rgt-out N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

   12. count-2 N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   13. distribute-lft1-in N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   17. *-lowering-*.f64 88.1%

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

3. Simplified 88.1%

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

5. Taylor expanded in x.im around inf

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

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

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

   4. cube-mult N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

   8. *-lowering-*.f64 58.4%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

7. Simplified 58.4%

   \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
8. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \frac{0 \cdot 0 - \left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)}{\color{blue}{0 + x.im \cdot \left(x.im \cdot x.im\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{0 - \left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{0 - \left(\left(x.im \cdot x.im\right) \cdot x.im\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{0 - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

   5. neg-sub0 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)\right)}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(x.im \cdot x.im\right) \cdot \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

   7. cube-unmult N/A

      \[\leadsto \frac{\mathsf{neg}\left({\left(x.im \cdot x.im\right)}^{3}\right)}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

   8. cube-neg N/A

      \[\leadsto \frac{{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)}^{3}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

   9. sub0-neg N/A

      \[\leadsto \frac{{\left(0 - x.im \cdot x.im\right)}^{3}}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

   10. sqr-pow N/A

       \[\leadsto \frac{{\left(0 - x.im \cdot x.im\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - x.im \cdot x.im\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

   11. pow-prod-down N/A

       \[\leadsto \frac{{\left(\left(0 - x.im \cdot x.im\right) \cdot \left(0 - x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

   12. sub0-neg N/A

       \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) \cdot \left(0 - x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)}}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

   13. sub0-neg N/A

       \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

   14. sqr-neg N/A

       \[\leadsto \frac{{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)}}{0 + x.im \cdot \left(x.im \cdot x.im\right)} \]

   15. unpow-prod-down N/A

       \[\leadsto \frac{{\left(x.im \cdot x.im\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(x.im \cdot x.im\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

   16. sqr-pow N/A

       \[\leadsto \frac{{\left(x.im \cdot x.im\right)}^{3}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

   17. unpow-prod-down N/A

       \[\leadsto \frac{{x.im}^{3} \cdot {x.im}^{3}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

   18. pow-sqr N/A

       \[\leadsto \frac{{x.im}^{\left(2 \cdot 3\right)}}{\color{blue}{0} + x.im \cdot \left(x.im \cdot x.im\right)} \]

   19. +-lft-identity N/A

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

   20. cube-unmult N/A

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

9. Applied egg-rr 19.4%

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

10. Final simplification 19.4%

    \[\leadsto x.im \cdot \left(x.im \cdot x.im\right) \]
11. Add Preprocessing

Developer Target 1: 91.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - 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_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - 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_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024130 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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