math.cube on complex, real part

Percentage Accurate: 83.0% → 99.8%

Time: 9.1s

Alternatives: 9

Speedup: 1.4×

• 43.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{+93}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m - x.re\_m\right) - x.im\_m \cdot \left(x.re\_m + x.re\_m \cdot 2\right)\right) + {x.re\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\_m \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2e+93)
    (+
     (*
      x.im_m
      (- (* x.re_m (- x.re_m x.re_m)) (* x.im_m (+ x.re_m (* x.re_m 2.0)))))
     (pow x.re_m 3.0))
    (* x.re_m (* x.re_m (+ x.re_m (* x.im_m -2.0)))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 2e+93) {
		tmp = (x_46_im_m * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im_m * (x_46_re_m + (x_46_re_m * 2.0))))) + pow(x_46_re_m, 3.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 2d+93) then
        tmp = (x_46im_m * ((x_46re_m * (x_46re_m - x_46re_m)) - (x_46im_m * (x_46re_m + (x_46re_m * 2.0d0))))) + (x_46re_m ** 3.0d0)
    else
        tmp = x_46re_m * (x_46re_m * (x_46re_m + (x_46im_m * (-2.0d0))))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 2e+93) {
		tmp = (x_46_im_m * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im_m * (x_46_re_m + (x_46_re_m * 2.0))))) + Math.pow(x_46_re_m, 3.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 2e+93:
		tmp = (x_46_im_m * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im_m * (x_46_re_m + (x_46_re_m * 2.0))))) + math.pow(x_46_re_m, 3.0)
	else:
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 2e+93)
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m * Float64(x_46_re_m - x_46_re_m)) - Float64(x_46_im_m * Float64(x_46_re_m + Float64(x_46_re_m * 2.0))))) + (x_46_re_m ^ 3.0));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_re_m + Float64(x_46_im_m * -2.0))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 2e+93)
		tmp = (x_46_im_m * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im_m * (x_46_re_m + (x_46_re_m * 2.0))))) + (x_46_re_m ^ 3.0);
	else
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2e+93], N[(N[(x$46$im$95$m * N[(N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$re$95$m + N[(x$46$re$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.00000000000000009e93

   1. Initial program 82.8%

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

      1. difference-of-squares 85.6%

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

      2. *-commutative 85.6%

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

   4. Applied egg-rr 85.6%

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

   5. Taylor expanded in x.im around 0 90.8%

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

   if 2.00000000000000009e93 < x.re

   1. Initial program 63.8%

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

      1. difference-of-squares 66.0%

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

      2. *-commutative 66.0%

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

   4. Applied egg-rr 66.0%

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

   5. Taylor expanded in x.re around 0 66.0%

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

      1. distribute-lft-in 57.4%

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

      2. mul-1-neg 57.4%

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

      3. sub-neg 57.4%

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

      4. associate-+r+ 57.4%

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

      5. unpow2 57.4%

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

      6. associate-*r* 57.4%

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

      7. distribute-rgt-in 61.7%

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

      8. +-commutative 61.7%

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

      9. mul-1-neg 61.7%

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

      10. sub-neg 61.7%

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

      11. distribute-lft-out 48.9%

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

      12. associate-*r* 48.9%

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

      13. associate-*r* 48.9%

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

      14. unpow2 48.9%

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

      15. distribute-rgt-out 63.8%

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

   7. Simplified 66.0%

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

   9. Applied egg-rr 89.4%

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

       1. fma-undefine 89.4%

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

       2. +-rgt-identity 89.4%

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

       3. *-commutative 89.4%

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

       4. associate-*l* 89.4%

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

   11. Simplified 89.4%

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

   12. Taylor expanded in x.im around 0 83.0%

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

       1. +-commutative 83.0%

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

       2. unpow2 83.0%

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

       3. associate-*r* 83.0%

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

       4. distribute-rgt-in 95.7%

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

       5. *-commutative 95.7%

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

   14. Simplified 95.7%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+93}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.re\right) - x.im \cdot \left(x.re + x.re \cdot 2\right)\right) + {x.re}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re + x.im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 4.7 \cdot 10^{-104}:\\ \;\;\;\;\left(x.im\_m \cdot -3\right) \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \mathbf{elif}\;x.re\_m \leq 2 \cdot 10^{+90}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m + x.im\_m\right)\right) - x.im\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\_m \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 4.7e-104)
    (* (* x.im_m -3.0) (* x.re_m x.im_m))
    (if (<= x.re_m 2e+90)
      (-
       (* x.re_m (* (- x.re_m x.im_m) (+ x.re_m x.im_m)))
       (* x.im_m (* x.re_m (* x.im_m 2.0))))
      (* x.re_m (* x.re_m (+ x.re_m (* x.im_m -2.0))))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 4.7e-104) {
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	} else if (x_46_re_m <= 2e+90) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_re_m * (x_46_im_m * 2.0)));
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 4.7d-104) then
        tmp = (x_46im_m * (-3.0d0)) * (x_46re_m * x_46im_m)
    else if (x_46re_m <= 2d+90) then
        tmp = (x_46re_m * ((x_46re_m - x_46im_m) * (x_46re_m + x_46im_m))) - (x_46im_m * (x_46re_m * (x_46im_m * 2.0d0)))
    else
        tmp = x_46re_m * (x_46re_m * (x_46re_m + (x_46im_m * (-2.0d0))))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 4.7e-104) {
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	} else if (x_46_re_m <= 2e+90) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_re_m * (x_46_im_m * 2.0)));
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 4.7e-104:
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m)
	elif x_46_re_m <= 2e+90:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_re_m * (x_46_im_m * 2.0)))
	else:
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 4.7e-104)
		tmp = Float64(Float64(x_46_im_m * -3.0) * Float64(x_46_re_m * x_46_im_m));
	elseif (x_46_re_m <= 2e+90)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + x_46_im_m))) - Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_im_m * 2.0))));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_re_m + Float64(x_46_im_m * -2.0))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 4.7e-104)
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	elseif (x_46_re_m <= 2e+90)
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_re_m * (x_46_im_m * 2.0)));
	else
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 4.7e-104], N[(N[(x$46$im$95$m * -3.0), $MachinePrecision] * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 2e+90], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 4.7e-104

   1. Initial program 79.3%

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

   3. Taylor expanded in x.im around inf 59.2%

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

   5. Applied egg-rr 25.1%

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

      1. exp-sum 25.1%

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

      2. rem-exp-log 35.9%

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

      3. *-commutative 35.9%

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

      4. rem-exp-log 70.2%

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

      5. *-commutative 70.2%

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

   7. Simplified 70.2%

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

   if 4.7e-104 < x.re < 1.99999999999999993e90

   1. Initial program 97.3%

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

      1. difference-of-squares 97.3%

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

      2. *-commutative 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x.re around 0 97.3%

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

      1. associate-*r* 97.3%

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

   7. Simplified 97.3%

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

   if 1.99999999999999993e90 < x.re

   1. Initial program 65.3%

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

      1. difference-of-squares 67.3%

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

      2. *-commutative 67.3%

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

   4. Applied egg-rr 67.3%

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

   5. Taylor expanded in x.re around 0 67.3%

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

      1. distribute-lft-in 59.2%

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

      2. mul-1-neg 59.2%

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

      3. sub-neg 59.2%

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

      4. associate-+r+ 59.2%

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

      5. unpow2 59.2%

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

      6. associate-*r* 59.2%

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

      7. distribute-rgt-in 63.2%

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

      8. +-commutative 63.2%

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

      9. mul-1-neg 63.2%

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

      10. sub-neg 63.2%

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

      11. distribute-lft-out 51.0%

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

      12. associate-*r* 51.0%

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

      13. associate-*r* 51.0%

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

      14. unpow2 51.0%

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

      15. distribute-rgt-out 65.3%

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

   7. Simplified 67.3%

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

   9. Applied egg-rr 89.8%

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

       1. fma-undefine 89.8%

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

       2. +-rgt-identity 89.8%

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

       3. *-commutative 89.8%

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

       4. associate-*l* 89.8%

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

   11. Simplified 89.8%

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

   12. Taylor expanded in x.im around 0 83.6%

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

       1. +-commutative 83.6%

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

       2. unpow2 83.6%

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

       3. associate-*r* 83.6%

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

       4. distribute-rgt-in 95.9%

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

       5. *-commutative 95.9%

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

   14. Simplified 95.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4.7 \cdot 10^{-104}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+90}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re + x.im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 10^{+91}:\\ \;\;\;\;\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\_m\right)\right) - x.im\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\_m \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1e+91)
    (-
     (* (- x.re_m x.im_m) (* x.re_m (+ x.re_m x.im_m)))
     (* x.im_m (* x.re_m (* x.im_m 2.0))))
    (* x.re_m (* x.re_m (+ x.re_m (* x.im_m -2.0)))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1e+91) {
		tmp = ((x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_re_m * (x_46_im_m * 2.0)));
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 1d+91) then
        tmp = ((x_46re_m - x_46im_m) * (x_46re_m * (x_46re_m + x_46im_m))) - (x_46im_m * (x_46re_m * (x_46im_m * 2.0d0)))
    else
        tmp = x_46re_m * (x_46re_m * (x_46re_m + (x_46im_m * (-2.0d0))))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1e+91) {
		tmp = ((x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_re_m * (x_46_im_m * 2.0)));
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 1e+91:
		tmp = ((x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_re_m * (x_46_im_m * 2.0)))
	else:
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 1e+91)
		tmp = Float64(Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m * Float64(x_46_re_m + x_46_im_m))) - Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_im_m * 2.0))));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_re_m + Float64(x_46_im_m * -2.0))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 1e+91)
		tmp = ((x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_re_m * (x_46_im_m * 2.0)));
	else
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1e+91], N[(N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.00000000000000008e91

   1. Initial program 82.6%

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

      1. difference-of-squares 85.5%

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

      2. *-commutative 85.5%

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

   4. Applied egg-rr 85.5%

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

   5. Taylor expanded in x.re around 0 85.5%

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

      1. associate-*r* 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in x.re around 0 84.1%

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

      1. distribute-lft-in 80.7%

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

      2. mul-1-neg 80.7%

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

      3. sub-neg 80.7%

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

      4. associate-+r+ 80.7%

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

      5. unpow2 80.7%

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

      6. associate-*r* 80.7%

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

      7. distribute-rgt-in 82.1%

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

      8. +-commutative 82.1%

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

      9. mul-1-neg 82.1%

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

      10. sub-neg 82.1%

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

      11. distribute-lft-out 77.8%

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

      12. associate-*r* 87.1%

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

      13. associate-*r* 87.1%

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

      14. unpow2 87.1%

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

      15. distribute-rgt-out 92.9%

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

   10. Simplified 94.9%

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

   if 1.00000000000000008e91 < x.re

   1. Initial program 65.3%

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

      1. difference-of-squares 67.3%

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

      2. *-commutative 67.3%

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

   4. Applied egg-rr 67.3%

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

   5. Taylor expanded in x.re around 0 67.3%

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

      1. distribute-lft-in 59.2%

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

      2. mul-1-neg 59.2%

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

      3. sub-neg 59.2%

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

      4. associate-+r+ 59.2%

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

      5. unpow2 59.2%

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

      6. associate-*r* 59.2%

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

      7. distribute-rgt-in 63.2%

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

      8. +-commutative 63.2%

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

      9. mul-1-neg 63.2%

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

      10. sub-neg 63.2%

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

      11. distribute-lft-out 51.0%

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

      12. associate-*r* 51.0%

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

      13. associate-*r* 51.0%

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

      14. unpow2 51.0%

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

      15. distribute-rgt-out 65.3%

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

   7. Simplified 67.3%

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

   9. Applied egg-rr 89.8%

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

       1. fma-undefine 89.8%

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

       2. +-rgt-identity 89.8%

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

       3. *-commutative 89.8%

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

       4. associate-*l* 89.8%

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

   11. Simplified 89.8%

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

   12. Taylor expanded in x.im around 0 83.6%

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

       1. +-commutative 83.6%

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

       2. unpow2 83.6%

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

       3. associate-*r* 83.6%

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

       4. distribute-rgt-in 95.9%

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

       5. *-commutative 95.9%

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

   14. Simplified 95.9%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 10^{+91}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re + x.im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.7 \cdot 10^{+48}:\\ \;\;\;\;\left(x.im\_m \cdot -3\right) \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\_m \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.7e+48)
    (* (* x.im_m -3.0) (* x.re_m x.im_m))
    (* x.re_m (* x.re_m (+ x.re_m (* x.im_m -2.0)))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 2.7e+48) {
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 2.7d+48) then
        tmp = (x_46im_m * (-3.0d0)) * (x_46re_m * x_46im_m)
    else
        tmp = x_46re_m * (x_46re_m * (x_46re_m + (x_46im_m * (-2.0d0))))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 2.7e+48) {
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 2.7e+48:
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m)
	else:
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 2.7e+48)
		tmp = Float64(Float64(x_46_im_m * -3.0) * Float64(x_46_re_m * x_46_im_m));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_re_m + Float64(x_46_im_m * -2.0))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 2.7e+48)
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	else
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.7e+48], N[(N[(x$46$im$95$m * -3.0), $MachinePrecision] * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.70000000000000004e48

   1. Initial program 82.2%

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

   3. Taylor expanded in x.im around inf 58.8%

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

   5. Applied egg-rr 21.0%

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

      1. exp-sum 21.0%

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

      2. rem-exp-log 33.1%

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

      3. *-commutative 33.1%

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

      4. rem-exp-log 68.4%

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

      5. *-commutative 68.4%

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

   7. Simplified 68.4%

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

   if 2.70000000000000004e48 < x.re

   1. Initial program 68.5%

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

      1. difference-of-squares 70.3%

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

      2. *-commutative 70.3%

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

   4. Applied egg-rr 70.3%

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

   5. Taylor expanded in x.re around 0 70.3%

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

      1. distribute-lft-in 61.1%

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

      2. mul-1-neg 61.1%

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

      3. sub-neg 61.1%

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

      4. associate-+r+ 61.1%

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

      5. unpow2 61.1%

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

      6. associate-*r* 61.1%

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

      7. distribute-rgt-in 64.8%

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

      8. +-commutative 64.8%

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

      9. mul-1-neg 64.8%

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

      10. sub-neg 64.8%

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

      11. distribute-lft-out 53.6%

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

      12. associate-*r* 53.6%

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

      13. associate-*r* 53.6%

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

      14. unpow2 53.6%

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

      15. distribute-rgt-out 68.5%

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

   7. Simplified 70.3%

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

   9. Applied egg-rr 87.0%

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

       1. fma-undefine 87.0%

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

       2. +-rgt-identity 87.0%

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

       3. *-commutative 87.0%

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

       4. associate-*l* 87.0%

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

   11. Simplified 87.0%

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

   12. Taylor expanded in x.im around 0 81.5%

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

       1. +-commutative 81.5%

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

       2. unpow2 81.5%

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

       3. associate-*r* 81.5%

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

       4. distribute-rgt-in 92.6%

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

       5. *-commutative 92.6%

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

   14. Simplified 92.6%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.7 \cdot 10^{+48}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re + x.im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 10^{-82}:\\ \;\;\;\;\left(x.im\_m \cdot -3\right) \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m + x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1e-82)
    (* (* x.im_m -3.0) (* x.re_m x.im_m))
    (* x.re_m (* (- x.re_m x.im_m) (+ x.re_m x.im_m))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1e-82) {
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 1d-82) then
        tmp = (x_46im_m * (-3.0d0)) * (x_46re_m * x_46im_m)
    else
        tmp = x_46re_m * ((x_46re_m - x_46im_m) * (x_46re_m + x_46im_m))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1e-82) {
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 1e-82:
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m)
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 1e-82)
		tmp = Float64(Float64(x_46_im_m * -3.0) * Float64(x_46_re_m * x_46_im_m));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + x_46_im_m)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 1e-82)
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1e-82], N[(N[(x$46$im$95$m * -3.0), $MachinePrecision] * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1e-82

   1. Initial program 79.8%

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

   3. Taylor expanded in x.im around inf 59.2%

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

   5. Applied egg-rr 24.5%

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

      1. exp-sum 24.5%

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

      2. rem-exp-log 35.7%

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

      3. *-commutative 35.7%

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

      4. rem-exp-log 69.9%

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

      5. *-commutative 69.9%

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

   7. Simplified 69.9%

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

   if 1e-82 < x.re

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. *-commutative 78.3%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \frac{\log 1}{\color{blue}{0}} \]

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \color{blue}{0}}{\log 1} \]

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Simplified 84.0%

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

      1. difference-of-squares 79.5%

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

      2. *-commutative 79.5%

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

   8. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.re - 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 10^{-82}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.7 \cdot 10^{-82}:\\ \;\;\;\;\left(x.im\_m \cdot -3\right) \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1.7e-82)
    (* (* x.im_m -3.0) (* x.re_m x.im_m))
    (* (- x.re_m x.im_m) (* x.re_m (+ x.re_m x.im_m))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1.7e-82) {
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	} else {
		tmp = (x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_re_m + x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 1.7d-82) then
        tmp = (x_46im_m * (-3.0d0)) * (x_46re_m * x_46im_m)
    else
        tmp = (x_46re_m - x_46im_m) * (x_46re_m * (x_46re_m + x_46im_m))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1.7e-82) {
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	} else {
		tmp = (x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_re_m + x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 1.7e-82:
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m)
	else:
		tmp = (x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_re_m + x_46_im_m))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 1.7e-82)
		tmp = Float64(Float64(x_46_im_m * -3.0) * Float64(x_46_re_m * x_46_im_m));
	else
		tmp = Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m * Float64(x_46_re_m + x_46_im_m)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 1.7e-82)
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	else
		tmp = (x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_re_m + x_46_im_m));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.7e-82], N[(N[(x$46$im$95$m * -3.0), $MachinePrecision] * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.69999999999999988e-82

   1. Initial program 79.8%

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

   3. Taylor expanded in x.im around inf 59.2%

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

   5. Applied egg-rr 24.5%

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

      1. exp-sum 24.5%

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

      2. rem-exp-log 35.7%

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

      3. *-commutative 35.7%

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

      4. rem-exp-log 69.9%

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

      5. *-commutative 69.9%

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

   7. Simplified 69.9%

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

   if 1.69999999999999988e-82 < x.re

   1. Initial program 78.3%

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

      1. difference-of-squares 79.5%

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

      2. *-commutative 79.5%

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

   4. Applied egg-rr 79.5%

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

   5. Taylor expanded in x.re around 0 79.5%

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

      1. distribute-lft-in 72.3%

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

      2. mul-1-neg 72.3%

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

      3. sub-neg 72.3%

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

      4. associate-+r+ 72.3%

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

      5. unpow2 72.3%

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

      6. associate-*r* 72.3%

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

      7. distribute-rgt-in 74.7%

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

      8. +-commutative 74.7%

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

      9. mul-1-neg 74.7%

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

      10. sub-neg 74.7%

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

      11. distribute-lft-out 67.4%

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

      12. associate-*r* 68.5%

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

      13. associate-*r* 68.5%

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

      14. unpow2 68.5%

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

      15. distribute-rgt-out 79.4%

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

   7. Simplified 80.6%

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

      1. *-commutative 78.3%

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

      2. *-commutative 78.3%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \frac{\log 1}{\color{blue}{0}} \]

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \color{blue}{0}}{\log 1} \]

      10. metadata-eval 0.0%

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

   9. Applied egg-rr 0.0%

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

   11. Simplified 91.4%

       \[\leadsto \left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right) - \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.7 \cdot 10^{-82}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 9.2 \cdot 10^{+135}:\\ \;\;\;\;-3 \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 9.2e+135)
    (* -3.0 (* x.re_m (* x.im_m x.im_m)))
    (* x.im_m (* x.re_m x.im_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 9.2e+135) {
		tmp = -3.0 * (x_46_re_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_im_m * (x_46_re_m * x_46_im_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 9.2d+135) then
        tmp = (-3.0d0) * (x_46re_m * (x_46im_m * x_46im_m))
    else
        tmp = x_46im_m * (x_46re_m * x_46im_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 9.2e+135) {
		tmp = -3.0 * (x_46_re_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_im_m * (x_46_re_m * x_46_im_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 9.2e+135:
		tmp = -3.0 * (x_46_re_m * (x_46_im_m * x_46_im_m))
	else:
		tmp = x_46_im_m * (x_46_re_m * x_46_im_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 9.2e+135)
		tmp = Float64(-3.0 * Float64(x_46_re_m * Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 9.2e+135)
		tmp = -3.0 * (x_46_re_m * (x_46_im_m * x_46_im_m));
	else
		tmp = x_46_im_m * (x_46_re_m * x_46_im_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 9.2e+135], N[(-3.0 * N[(x$46$re$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 9.2000000000000005e135

   1. Initial program 83.2%

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

   3. Simplified 80.5%

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

      1. associate-*r* 80.5%

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

      2. associate-*l* 80.4%

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

      3. +-commutative 80.4%

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

      4. associate-*r* 89.5%

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

      5. associate-*r* 89.6%

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

      6. fma-define 91.0%

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

   6. Applied egg-rr 91.0%

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

   7. Taylor expanded in x.re around 0 57.5%

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

      1. unpow2 57.5%

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

   9. Applied egg-rr 57.5%

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

   if 9.2000000000000005e135 < x.re

   1. Initial program 59.5%

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

   3. Taylor expanded in x.im around inf 5.3%

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

   5. Applied egg-rr 0.0%

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

      1. exp-sum 0.0%

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

      2. rem-exp-log 2.7%

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

      3. *-commutative 2.7%

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

      4. rem-exp-log 5.3%

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

      5. *-commutative 5.3%

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

   7. Simplified 5.3%

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

      1. associate-*l* 5.3%

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

      2. *-commutative 5.3%

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

      3. associate-*r* 5.3%

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

      4. *-commutative 5.3%

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

      5. associate-*r* 5.3%

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

      6. metadata-eval 5.3%

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

      7. distribute-rgt-out-- 5.3%

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

      8. pow1 5.3%

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

   9. Applied egg-rr 40.0%

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

       1. unpow1 40.0%

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

   11. Simplified 40.0%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 9.2 \cdot 10^{+135}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 9.2 \cdot 10^{+135}:\\ \;\;\;\;\left(x.im\_m \cdot -3\right) \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 9.2e+135)
    (* (* x.im_m -3.0) (* x.re_m x.im_m))
    (* x.im_m (* x.re_m x.im_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 9.2e+135) {
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	} else {
		tmp = x_46_im_m * (x_46_re_m * x_46_im_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 9.2d+135) then
        tmp = (x_46im_m * (-3.0d0)) * (x_46re_m * x_46im_m)
    else
        tmp = x_46im_m * (x_46re_m * x_46im_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 9.2e+135) {
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	} else {
		tmp = x_46_im_m * (x_46_re_m * x_46_im_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 9.2e+135:
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m)
	else:
		tmp = x_46_im_m * (x_46_re_m * x_46_im_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 9.2e+135)
		tmp = Float64(Float64(x_46_im_m * -3.0) * Float64(x_46_re_m * x_46_im_m));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 9.2e+135)
		tmp = (x_46_im_m * -3.0) * (x_46_re_m * x_46_im_m);
	else
		tmp = x_46_im_m * (x_46_re_m * x_46_im_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 9.2e+135], N[(N[(x$46$im$95$m * -3.0), $MachinePrecision] * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 9.2000000000000005e135

   1. Initial program 83.2%

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

   3. Taylor expanded in x.im around inf 57.6%

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

   5. Applied egg-rr 19.8%

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

      1. exp-sum 19.8%

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

      2. rem-exp-log 32.4%

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

      3. *-commutative 32.4%

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

      4. rem-exp-log 66.6%

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

      5. *-commutative 66.6%

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

   7. Simplified 66.6%

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

   if 9.2000000000000005e135 < x.re

   1. Initial program 59.5%

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

   3. Taylor expanded in x.im around inf 5.3%

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

   5. Applied egg-rr 0.0%

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

      1. exp-sum 0.0%

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

      2. rem-exp-log 2.7%

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

      3. *-commutative 2.7%

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

      4. rem-exp-log 5.3%

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

      5. *-commutative 5.3%

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

   7. Simplified 5.3%

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

      1. associate-*l* 5.3%

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

      2. *-commutative 5.3%

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

      3. associate-*r* 5.3%

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

      4. *-commutative 5.3%

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

      5. associate-*r* 5.3%

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

      6. metadata-eval 5.3%

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

      7. distribute-rgt-out-- 5.3%

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

      8. pow1 5.3%

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

   9. Applied egg-rr 40.0%

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

       1. unpow1 40.0%

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

   11. Simplified 40.0%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 9.2 \cdot 10^{+135}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 24.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right) \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (* x.re_s (* x.im_m (* x.re_m x.im_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_im_m * (x_46_re_m * x_46_im_m));
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * (x_46im_m * (x_46re_m * x_46im_m))
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_im_m * (x_46_re_m * x_46_im_m));
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * (x_46_im_m * (x_46_re_m * x_46_im_m))

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_m)))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * (x_46_im_m * (x_46_re_m * x_46_im_m));
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in x.im around inf 49.0%

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

5. Applied egg-rr 16.6%

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

   1. exp-sum 16.6%

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

   2. rem-exp-log 27.5%

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

   3. *-commutative 27.5%

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

   4. rem-exp-log 56.6%

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

   5. *-commutative 56.6%

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

7. Simplified 56.6%

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

   1. associate-*l* 56.6%

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

   2. *-commutative 56.6%

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

   3. associate-*r* 56.6%

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

   4. *-commutative 56.6%

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

   5. associate-*r* 56.6%

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

   6. metadata-eval 56.6%

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

   7. distribute-rgt-out-- 56.6%

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

   8. pow1 56.6%

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

9. Applied egg-rr 26.9%

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

    1. unpow1 26.9%

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

11. Simplified 26.9%

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

12. Final simplification 26.9%

    \[\leadsto x.im \cdot \left(x.re \cdot x.im\right) \]
13. Add Preprocessing

Developer target: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024080 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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