math.cube on complex, real part

Percentage Accurate: 82.6% → 99.8%

Time: 7.3s

Alternatives: 7

Speedup: 0.5×

• 44.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.6% 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, 0.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(x.im \cdot \left(x.re\_m \cdot \left(x.re\_m - x.re\_m\right) - x.re\_m \cdot x.im\right) + {x.re\_m}^{3}\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5e+102)
    (-
     (+
      (* x.im (- (* x.re_m (- x.re_m x.re_m)) (* x.re_m x.im)))
      (pow x.re_m 3.0))
     (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
    (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))))

C

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) {
	double tmp;
	if (x_46_re_m <= 5e+102) {
		tmp = ((x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_re_m * x_46_im))) + pow(x_46_re_m, 3.0)) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 5d+102) then
        tmp = ((x_46im * ((x_46re_m * (x_46re_m - x_46re_m)) - (x_46re_m * x_46im))) + (x_46re_m ** 3.0d0)) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double tmp;
	if (x_46_re_m <= 5e+102) {
		tmp = ((x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_re_m * x_46_im))) + Math.pow(x_46_re_m, 3.0)) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

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):
	tmp = 0
	if x_46_re_m <= 5e+102:
		tmp = ((x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_re_m * x_46_im))) + math.pow(x_46_re_m, 3.0)) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

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)
	tmp = 0.0
	if (x_46_re_m <= 5e+102)
		tmp = Float64(Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * Float64(x_46_re_m - x_46_re_m)) - Float64(x_46_re_m * x_46_im))) + (x_46_re_m ^ 3.0)) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	tmp = 0.0;
	if (x_46_re_m <= 5e+102)
		tmp = ((x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_re_m * x_46_im))) + (x_46_re_m ^ 3.0)) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 5e+102], N[(N[(N[(x$46$im * N[(N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 5e102

   1. Initial program 85.9%

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

      1. difference-of-squares 88.2%

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

      2. *-commutative 88.2%

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

   4. Applied egg-rr 88.2%

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

   5. Taylor expanded in x.im around 0 89.8%

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

   if 5e102 < x.re

   1. Initial program 73.7%

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

      1. difference-of-squares 86.8%

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

      2. *-commutative 86.8%

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

   4. Applied egg-rr 86.8%

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

      1. associate-*l* 86.8%

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

      2. fma-neg 86.8%

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

      3. *-commutative 86.8%

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

      4. *-commutative 86.8%

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

      5. distribute-rgt-neg-in 86.8%

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

      6. *-commutative 86.8%

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

      7. flip-+ 0.0%

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

      8. *-commutative 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. *-commutative 0.0%

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

      12. distribute-neg-frac2 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x.re around 0 89.5%

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

       1. distribute-lft-in 71.1%

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

       2. *-commutative 71.1%

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

       3. neg-mul-1 71.1%

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

       4. sub-neg 71.1%

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

       5. associate-+r+ 71.1%

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

       6. +-commutative 71.1%

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

       7. mul-1-neg 71.1%

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

       8. unpow2 71.1%

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

       9. distribute-rgt-neg-in 71.1%

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

       10. distribute-lft-in 89.5%

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

       11. sub-neg 89.5%

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

       12. +-commutative 89.5%

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

       13. distribute-rgt-out 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot \left(x.re - x.re\right) - x.re \cdot x.im\right) + {x.re}^{3}\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{+101}:\\ \;\;\;\;{x.re\_m}^{3} + x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2e+101)
    (+ (pow x.re_m 3.0) (* x.im (* x.re_m (* x.im -3.0))))
    (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))))

C

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) {
	double tmp;
	if (x_46_re_m <= 2e+101) {
		tmp = pow(x_46_re_m, 3.0) + (x_46_im * (x_46_re_m * (x_46_im * -3.0)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2d+101) then
        tmp = (x_46re_m ** 3.0d0) + (x_46im * (x_46re_m * (x_46im * (-3.0d0))))
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double tmp;
	if (x_46_re_m <= 2e+101) {
		tmp = Math.pow(x_46_re_m, 3.0) + (x_46_im * (x_46_re_m * (x_46_im * -3.0)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

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):
	tmp = 0
	if x_46_re_m <= 2e+101:
		tmp = math.pow(x_46_re_m, 3.0) + (x_46_im * (x_46_re_m * (x_46_im * -3.0)))
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

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)
	tmp = 0.0
	if (x_46_re_m <= 2e+101)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	tmp = 0.0;
	if (x_46_re_m <= 2e+101)
		tmp = (x_46_re_m ^ 3.0) + (x_46_im * (x_46_re_m * (x_46_im * -3.0)));
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2e+101], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2e101

   1. Initial program 85.9%

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

      1. difference-of-squares 88.2%

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

      2. *-commutative 88.2%

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

   4. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

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

      2. distribute-rgt-in 85.9%

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

      3. distribute-rgt-in 82.7%

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in x.im around 0 83.8%

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

      1. fma-define 83.8%

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

      2. +-commutative 83.8%

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

      3. associate-+r+ 83.8%

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

      4. distribute-lft1-in 83.8%

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

      5. metadata-eval 83.8%

         \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{0} \cdot {x.re}^{2} + x.im \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right), {x.re}^{3}\right) \]

      6. mul0-lft 90.7%

         \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{0} + x.im \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right), {x.re}^{3}\right) \]

      7. +-lft-identity 90.7%

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

      8. *-commutative 90.7%

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

      9. distribute-rgt-out-- 90.7%

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

      10. metadata-eval 90.7%

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

      11. metadata-eval 90.7%

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

      12. associate-*l* 90.7%

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

      13. metadata-eval 90.7%

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

   9. Simplified 90.7%

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

       1. fma-undefine 89.7%

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

       2. *-commutative 89.7%

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

   11. Applied egg-rr 89.7%

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

   if 2e101 < x.re

   1. Initial program 73.7%

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

      1. difference-of-squares 86.8%

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

      2. *-commutative 86.8%

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

   4. Applied egg-rr 86.8%

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

      1. associate-*l* 86.8%

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

      2. fma-neg 86.8%

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

      3. *-commutative 86.8%

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

      4. *-commutative 86.8%

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

      5. distribute-rgt-neg-in 86.8%

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

      6. *-commutative 86.8%

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

      7. flip-+ 0.0%

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

      8. *-commutative 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. *-commutative 0.0%

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

      12. distribute-neg-frac2 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x.re around 0 89.5%

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

       1. distribute-lft-in 71.1%

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

       2. *-commutative 71.1%

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

       3. neg-mul-1 71.1%

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

       4. sub-neg 71.1%

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

       5. associate-+r+ 71.1%

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

       6. +-commutative 71.1%

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

       7. mul-1-neg 71.1%

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

       8. unpow2 71.1%

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

       9. distribute-rgt-neg-in 71.1%

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

       10. distribute-lft-in 89.5%

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

       11. sub-neg 89.5%

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

       12. +-commutative 89.5%

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

       13. distribute-rgt-out 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+101}:\\ \;\;\;\;{x.re}^{3} + x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - t\_0 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(x.re\_m \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\right)\right) + \left(x.re\_m \cdot x.im\right) \cdot \left(x.re\_m - x.im\right)\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (let* ((t_0 (* x.im (+ (* x.re_m x.im) (* x.re_m x.im)))))
   (*
    x.re_s
    (if (<= (- (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))) t_0) -5e-311)
      (-
       (+
        (* x.re_m (* x.re_m (- x.re_m x.im)))
        (* (* x.re_m x.im) (- x.re_m x.im)))
       t_0)
      (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))))))

C

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) {
	double t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-311) {
		tmp = ((x_46_re_m * (x_46_re_m * (x_46_re_m - x_46_im))) + ((x_46_re_m * x_46_im) * (x_46_re_m - x_46_im))) - t_0;
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im))
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - t_0) <= (-5d-311)) then
        tmp = ((x_46re_m * (x_46re_m * (x_46re_m - x_46im))) + ((x_46re_m * x_46im) * (x_46re_m - x_46im))) - t_0
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-311) {
		tmp = ((x_46_re_m * (x_46_re_m * (x_46_re_m - x_46_im))) + ((x_46_re_m * x_46_im) * (x_46_re_m - x_46_im))) - t_0;
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

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):
	t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im))
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-311:
		tmp = ((x_46_re_m * (x_46_re_m * (x_46_re_m - x_46_im))) + ((x_46_re_m * x_46_im) * (x_46_re_m - x_46_im))) - t_0
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

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)
	t_0 = Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - t_0) <= -5e-311)
		tmp = Float64(Float64(Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im))) + Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_re_m - x_46_im))) - t_0);
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-311)
		tmp = ((x_46_re_m * (x_46_re_m * (x_46_re_m - x_46_im))) + ((x_46_re_m * x_46_im) * (x_46_re_m - x_46_im))) - t_0;
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := Block[{t$95$0 = N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], -5e-311], N[(N[(N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -5.00000000000023e-311

   1. Initial program 97.8%

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

      1. difference-of-squares 97.8%

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

      2. *-commutative 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. distribute-rgt-in 96.7%

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

      3. distribute-rgt-in 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in x.im around 0 88.2%

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

      1. mul-1-neg 88.2%

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

      2. distribute-lft-neg-out 88.2%

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

      3. +-commutative 88.2%

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

      4. unpow2 88.2%

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

      5. distribute-rgt-in 88.2%

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

      6. sub-neg 88.2%

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

      7. associate-*l* 93.4%

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

      8. *-commutative 93.4%

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

   9. Simplified 93.4%

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

   if -5.00000000000023e-311 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 76.0%

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

      1. difference-of-squares 82.3%

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

      2. *-commutative 82.3%

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

   4. Applied egg-rr 82.3%

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

      1. associate-*l* 88.6%

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

      2. fma-neg 88.6%

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

      3. *-commutative 88.6%

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

      4. *-commutative 88.6%

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

      5. distribute-rgt-neg-in 88.6%

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

      6. *-commutative 88.6%

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

      7. flip-+ 0.0%

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

      8. *-commutative 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. *-commutative 0.0%

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

      12. distribute-neg-frac2 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in x.re around 0 71.9%

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

       1. distribute-lft-in 63.9%

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

       2. *-commutative 63.9%

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

       3. neg-mul-1 63.9%

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

       4. sub-neg 63.9%

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

       5. associate-+r+ 63.9%

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

       6. +-commutative 63.9%

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

       7. mul-1-neg 63.9%

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

       8. unpow2 63.9%

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

       9. distribute-rgt-neg-in 63.9%

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

       10. distribute-lft-in 70.1%

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

       11. sub-neg 70.1%

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

       12. +-commutative 70.1%

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

       13. distribute-rgt-out 78.8%

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

   11. Simplified 78.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(x.re \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - t\_0 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(x.re\_m \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\right)\right) + x.re\_m \cdot \left(x.im \cdot \left(x.re\_m - x.im\right)\right)\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (let* ((t_0 (* x.im (+ (* x.re_m x.im) (* x.re_m x.im)))))
   (*
    x.re_s
    (if (<= (- (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))) t_0) -5e-311)
      (-
       (+
        (* x.re_m (* x.re_m (- x.re_m x.im)))
        (* x.re_m (* x.im (- x.re_m x.im))))
       t_0)
      (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))))))

C

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) {
	double t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-311) {
		tmp = ((x_46_re_m * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_re_m * (x_46_im * (x_46_re_m - x_46_im)))) - t_0;
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im))
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - t_0) <= (-5d-311)) then
        tmp = ((x_46re_m * (x_46re_m * (x_46re_m - x_46im))) + (x_46re_m * (x_46im * (x_46re_m - x_46im)))) - t_0
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-311) {
		tmp = ((x_46_re_m * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_re_m * (x_46_im * (x_46_re_m - x_46_im)))) - t_0;
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

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):
	t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im))
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-311:
		tmp = ((x_46_re_m * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_re_m * (x_46_im * (x_46_re_m - x_46_im)))) - t_0
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

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)
	t_0 = Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - t_0) <= -5e-311)
		tmp = Float64(Float64(Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im))) + Float64(x_46_re_m * Float64(x_46_im * Float64(x_46_re_m - x_46_im)))) - t_0);
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-311)
		tmp = ((x_46_re_m * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_re_m * (x_46_im * (x_46_re_m - x_46_im)))) - t_0;
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := Block[{t$95$0 = N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], -5e-311], N[(N[(N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$im * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -5.00000000000023e-311

   1. Initial program 97.8%

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

      1. difference-of-squares 97.8%

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

      2. *-commutative 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. distribute-rgt-in 96.7%

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

      3. distribute-rgt-in 91.5%

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

   6. Applied egg-rr 91.5%

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

   if -5.00000000000023e-311 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 76.0%

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

      1. difference-of-squares 82.3%

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

      2. *-commutative 82.3%

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

   4. Applied egg-rr 82.3%

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

      1. associate-*l* 88.6%

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

      2. fma-neg 88.6%

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

      3. *-commutative 88.6%

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

      4. *-commutative 88.6%

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

      5. distribute-rgt-neg-in 88.6%

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

      6. *-commutative 88.6%

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

      7. flip-+ 0.0%

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

      8. *-commutative 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. *-commutative 0.0%

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

      12. distribute-neg-frac2 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in x.re around 0 71.9%

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

       1. distribute-lft-in 63.9%

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

       2. *-commutative 63.9%

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

       3. neg-mul-1 63.9%

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

       4. sub-neg 63.9%

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

       5. associate-+r+ 63.9%

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

       6. +-commutative 63.9%

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

       7. mul-1-neg 63.9%

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

       8. unpow2 63.9%

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

       9. distribute-rgt-neg-in 63.9%

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

       10. distribute-lft-in 70.1%

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

       11. sub-neg 70.1%

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

       12. +-commutative 70.1%

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

       13. distribute-rgt-out 78.8%

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

   11. Simplified 78.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(x.re \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + x.re \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_0 - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (let* ((t_0 (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))))
   (*
    x.re_s
    (if (<=
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
         -5e-311)
      (- t_0 (* x.im (* (* x.re_m x.im) 2.0)))
      t_0))))

C

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) {
	double t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-311) {
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-5d-311)) then
        tmp = t_0 - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else
        tmp = t_0
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-311) {
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Python

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):
	t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-311:
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0))
	else:
		tmp = t_0
	return x_46_re_s * tmp

Julia

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)
	t_0 = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)))
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -5e-311)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * 2.0)));
	else
		tmp = t_0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-311)
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-311], N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -5.00000000000023e-311

   1. Initial program 97.8%

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

      1. difference-of-squares 97.8%

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

      2. *-commutative 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. distribute-rgt-in 96.7%

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

      3. distribute-rgt-in 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in x.im around 0 88.2%

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

      1. mul-1-neg 88.2%

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

      2. distribute-lft-neg-out 88.2%

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

      3. +-commutative 88.2%

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

      4. unpow2 88.2%

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

      5. distribute-rgt-in 88.2%

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

      6. sub-neg 88.2%

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

      7. associate-*l* 93.4%

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

      8. *-commutative 93.4%

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

   9. Simplified 93.4%

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

       1. associate--l+ 93.4%

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

       2. *-commutative 93.4%

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

       3. associate-*l* 91.5%

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

       4. *-commutative 91.5%

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

       5. fma-define 91.5%

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

       6. *-commutative 91.5%

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

   11. Applied egg-rr 91.5%

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

       1. associate-+r- 91.5%

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

       2. distribute-lft-in 96.7%

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

       3. distribute-rgt-in 97.8%

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

       4. fma-undefine 97.8%

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

       5. count-2 97.8%

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

   13. Simplified 97.8%

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

   if -5.00000000000023e-311 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 76.0%

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

      1. difference-of-squares 82.3%

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

      2. *-commutative 82.3%

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

   4. Applied egg-rr 82.3%

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

      1. associate-*l* 88.6%

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

      2. fma-neg 88.6%

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

      3. *-commutative 88.6%

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

      4. *-commutative 88.6%

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

      5. distribute-rgt-neg-in 88.6%

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

      6. *-commutative 88.6%

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

      7. flip-+ 0.0%

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

      8. *-commutative 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. *-commutative 0.0%

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

      12. distribute-neg-frac2 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in x.re around 0 71.9%

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

       1. distribute-lft-in 63.9%

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

       2. *-commutative 63.9%

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

       3. neg-mul-1 63.9%

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

       4. sub-neg 63.9%

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

       5. associate-+r+ 63.9%

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

       6. +-commutative 63.9%

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

       7. mul-1-neg 63.9%

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

       8. unpow2 63.9%

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

       9. distribute-rgt-neg-in 63.9%

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

       10. distribute-lft-in 70.1%

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

       11. sub-neg 70.1%

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

       12. +-commutative 70.1%

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

       13. distribute-rgt-out 78.8%

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

   11. Simplified 78.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 2.1× speedup

Math

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

FPCore

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)
 :precision binary64
 (* x.re_s (* (- x.re_m x.im) (* x.re_m (+ x.re_m x.im)))))

C

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) {
	return x_46_re_s * ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im)));
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((x_46re_m - x_46im) * (x_46re_m * (x_46re_m + x_46im)))
end function

Java

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) {
	return x_46_re_s * ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im)));
}

Python

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):
	return x_46_re_s * ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im)))

Julia

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)
	return Float64(x_46_re_s * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m + x_46_im))))
end

MATLAB

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)
	tmp = x_46_re_s * ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im)));
end

Wolfram

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_] := N[(x$46$re$95$s * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

   1. difference-of-squares 88.0%

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

   2. *-commutative 88.0%

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

4. Applied egg-rr 88.0%

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

   1. associate-*l* 92.7%

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

   2. fma-neg 92.7%

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

   3. *-commutative 92.7%

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

   4. *-commutative 92.7%

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

   5. distribute-rgt-neg-in 92.7%

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

   6. *-commutative 92.7%

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

   7. flip-+ 0.0%

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

   8. *-commutative 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. *-commutative 0.0%

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

   12. distribute-neg-frac2 0.0%

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

6. Applied egg-rr 0.0%

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

8. Simplified 76.8%

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

   1. fma-undefine 76.8%

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

   2. mul0-rgt 76.8%

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

10. Applied egg-rr 76.8%

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

11. Final simplification 76.8%

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

Alternative 7: 78.1% accurate, 2.1× speedup

Math

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

FPCore

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)
 :precision binary64
 (* x.re_s (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))))

C

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) {
	return x_46_re_s * (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im)));
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im)))
end function

Java

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) {
	return x_46_re_s * (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im)));
}

Python

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):
	return x_46_re_s * (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im)))

Julia

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)
	return Float64(x_46_re_s * Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))))
end

MATLAB

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)
	tmp = x_46_re_s * (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im)));
end

Wolfram

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_] := N[(x$46$re$95$s * N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

   1. difference-of-squares 88.0%

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

   2. *-commutative 88.0%

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

4. Applied egg-rr 88.0%

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

   1. associate-*l* 92.7%

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

   2. fma-neg 92.7%

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

   3. *-commutative 92.7%

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

   4. *-commutative 92.7%

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

   5. distribute-rgt-neg-in 92.7%

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

   6. *-commutative 92.7%

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

   7. flip-+ 0.0%

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

   8. *-commutative 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. *-commutative 0.0%

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

   12. distribute-neg-frac2 0.0%

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

6. Applied egg-rr 0.0%

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

8. Simplified 76.8%

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

9. Taylor expanded in x.re around 0 71.9%

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

    1. distribute-lft-in 66.1%

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

    2. *-commutative 66.1%

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

    3. neg-mul-1 66.1%

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

    4. sub-neg 66.1%

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

    5. associate-+r+ 66.1%

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

    6. +-commutative 66.1%

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

    7. mul-1-neg 66.1%

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

    8. unpow2 66.1%

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

    9. distribute-rgt-neg-in 66.1%

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

    10. distribute-lft-in 70.4%

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

    11. sub-neg 70.4%

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

    12. +-commutative 70.4%

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

    13. distribute-rgt-out 76.2%

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

11. Simplified 76.2%

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

Developer target: 87.7% 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 2024111 
(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)))