math.cube on complex, imaginary part

Percentage Accurate: 82.9% → 96.4%

Time: 7.7s

Alternatives: 6

Speedup: 1.2×

• 44.6% 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: 82.9% 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: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 3.4 \cdot 10^{+145}:\\ \;\;\;\;x.im \cdot \left(\left(x.re\_m \cdot x.re\_m\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 3.4e+145)
   (* x.im (- (* (* x.re_m x.re_m) 3.0) (* x.im x.im)))
   (* 3.0 (* x.re_m (* x.re_m 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_re_m <= 3.4e+145) {
		tmp = x_46_im * (((x_46_re_m * x_46_re_m) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * 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_46re_m <= 3.4d+145) then
        tmp = x_46im * (((x_46re_m * x_46re_m) * 3.0d0) - (x_46im * x_46im))
    else
        tmp = 3.0d0 * (x_46re_m * (x_46re_m * 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_re_m <= 3.4e+145) {
		tmp = x_46_im * (((x_46_re_m * x_46_re_m) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * 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_re_m <= 3.4e+145:
		tmp = x_46_im * (((x_46_re_m * x_46_re_m) * 3.0) - (x_46_im * x_46_im))
	else:
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * 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_re_m <= 3.4e+145)
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re_m * x_46_re_m) * 3.0) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re_m * Float64(x_46_re_m * 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_re_m <= 3.4e+145)
		tmp = x_46_im * (((x_46_re_m * x_46_re_m) * 3.0) - (x_46_im * x_46_im));
	else
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * 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$46$re$95$m, 3.4e+145], N[(x$46$im * N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.3999999999999999e145

   1. Initial program 82.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. count-2 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 92.3%

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

   3. Simplified 92.3%

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

   if 3.3999999999999999e145 < x.re

   1. Initial program 51.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. count-2 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 51.9%

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

   3. Simplified 51.9%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.4%

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

   7. Simplified 69.4%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 92.3%

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

   9. Applied egg-rr 92.3%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.4 \cdot 10^{+145}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.7 \cdot 10^{+67}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 1.7e+67)
   (* x.im (- 0.0 (* x.im x.im)))
   (* x.re_m (* x.re_m (* x.im 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 <= 1.7e+67) {
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 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 <= 1.7d+67) then
        tmp = x_46im * (0.0d0 - (x_46im * x_46im))
    else
        tmp = x_46re_m * (x_46re_m * (x_46im * 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 <= 1.7e+67) {
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 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 <= 1.7e+67:
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im))
	else:
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 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 <= 1.7e+67)
		tmp = Float64(x_46_im * Float64(0.0 - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im * 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 <= 1.7e+67)
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	else
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 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, 1.7e+67], N[(x$46$im * N[(0.0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.7000000000000001e67

   1. Initial program 85.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. count-2 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 91.4%

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

   3. Simplified 91.4%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 69.7%

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

   7. Simplified 69.7%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 69.7%

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

   9. Applied egg-rr 69.7%

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

   if 1.7000000000000001e67 < x.re

   1. Initial program 55.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. count-2 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 69.4%

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

   3. Simplified 69.4%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 65.6%

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

   7. Simplified 65.6%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 80.3%

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

   9. Applied egg-rr 80.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.7 \cdot 10^{+67}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.22 \cdot 10^{+67}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(3 \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 1.22e+67)
   (* x.im (- 0.0 (* x.im x.im)))
   (* x.re_m (* 3.0 (* x.re_m 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_re_m <= 1.22e+67) {
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (3.0 * (x_46_re_m * 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_46re_m <= 1.22d+67) then
        tmp = x_46im * (0.0d0 - (x_46im * x_46im))
    else
        tmp = x_46re_m * (3.0d0 * (x_46re_m * 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_re_m <= 1.22e+67) {
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (3.0 * (x_46_re_m * 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_re_m <= 1.22e+67:
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im))
	else:
		tmp = x_46_re_m * (3.0 * (x_46_re_m * 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_re_m <= 1.22e+67)
		tmp = Float64(x_46_im * Float64(0.0 - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(3.0 * Float64(x_46_re_m * 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_re_m <= 1.22e+67)
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	else
		tmp = x_46_re_m * (3.0 * (x_46_re_m * 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$46$re$95$m, 1.22e+67], N[(x$46$im * N[(0.0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(3.0 * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.22000000000000004e67

   1. Initial program 85.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. count-2 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 91.4%

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

   3. Simplified 91.4%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 69.7%

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

   7. Simplified 69.7%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 69.7%

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

   9. Applied egg-rr 69.7%

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

   if 1.22000000000000004e67 < x.re

   1. Initial program 55.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. count-2 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 69.4%

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

   3. Simplified 69.4%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 65.6%

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

   7. Simplified 65.6%

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

      1. associate-*r* 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 80.2%

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

   9. Applied egg-rr 80.2%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.22 \cdot 10^{+67}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.24 \cdot 10^{+67}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 1.24e+67)
   (* x.im (- 0.0 (* x.im x.im)))
   (* 3.0 (* x.re_m (* x.re_m 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_re_m <= 1.24e+67) {
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * 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_46re_m <= 1.24d+67) then
        tmp = x_46im * (0.0d0 - (x_46im * x_46im))
    else
        tmp = 3.0d0 * (x_46re_m * (x_46re_m * 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_re_m <= 1.24e+67) {
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * 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_re_m <= 1.24e+67:
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im))
	else:
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * 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_re_m <= 1.24e+67)
		tmp = Float64(x_46_im * Float64(0.0 - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re_m * Float64(x_46_re_m * 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_re_m <= 1.24e+67)
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	else
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * 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$46$re$95$m, 1.24e+67], N[(x$46$im * N[(0.0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.24000000000000007e67

   1. Initial program 85.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. count-2 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 91.4%

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

   3. Simplified 91.4%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 69.7%

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

   7. Simplified 69.7%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 69.7%

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

   9. Applied egg-rr 69.7%

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

   if 1.24000000000000007e67 < x.re

   1. Initial program 55.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. count-2 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 69.4%

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

   3. Simplified 69.4%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 65.6%

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

   7. Simplified 65.6%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 80.2%

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

   9. Applied egg-rr 80.2%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.24 \cdot 10^{+67}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.85 \cdot 10^{+67}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 1.85e+67)
   (* x.im (- 0.0 (* x.im x.im)))
   (* 3.0 (* x.im (* x.re_m x.re_m)))))

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 <= 1.85e+67) {
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re_m * x_46_re_m));
	}
	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 <= 1.85d+67) then
        tmp = x_46im * (0.0d0 - (x_46im * x_46im))
    else
        tmp = 3.0d0 * (x_46im * (x_46re_m * x_46re_m))
    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 <= 1.85e+67) {
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re_m * x_46_re_m));
	}
	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 <= 1.85e+67:
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im))
	else:
		tmp = 3.0 * (x_46_im * (x_46_re_m * x_46_re_m))
	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 <= 1.85e+67)
		tmp = Float64(x_46_im * Float64(0.0 - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_re_m)));
	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 <= 1.85e+67)
		tmp = x_46_im * (0.0 - (x_46_im * x_46_im));
	else
		tmp = 3.0 * (x_46_im * (x_46_re_m * x_46_re_m));
	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, 1.85e+67], N[(x$46$im * N[(0.0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.8499999999999999e67

   1. Initial program 85.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. count-2 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 91.4%

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

   3. Simplified 91.4%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 69.7%

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

   7. Simplified 69.7%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 69.7%

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

   9. Applied egg-rr 69.7%

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

   if 1.8499999999999999e67 < x.re

   1. Initial program 55.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. count-2 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 69.4%

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

   3. Simplified 69.4%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 65.6%

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

   7. Simplified 65.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.85 \cdot 10^{+67}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im \cdot \left(0 - x.im \cdot x.im\right) \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im) :precision binary64 (* x.im (- 0.0 (* 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 * (0.0 - (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 * (0.0d0 - (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 * (0.0 - (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 * (0.0 - (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(0.0 - 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 * (0.0 - (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[(0.0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
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. count-2 N/A

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

   11. associate-*l* N/A

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

   12. distribute-lft1-in N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

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

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

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

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

   17. *-lowering-*.f64 86.0%

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

3. Simplified 86.0%

   \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
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. unpow3 N/A

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

   3. unpow2 N/A

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. neg-sub0 N/A

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

   9. --lowering--.f64 57.5%

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

7. Simplified 57.5%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 57.5%

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

9. Applied egg-rr 57.5%

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

10. Final simplification 57.5%

    \[\leadsto x.im \cdot \left(0 - x.im \cdot x.im\right) \]
11. Add Preprocessing

Developer Target 1: 91.5% 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 2024145 
(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)))