math.cube on complex, real part

Percentage Accurate: 82.0% → 99.8%

Time: 11.4s

Alternatives: 6

Speedup: 0.7×

• 44.2% 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: 82.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+208}:\\ \;\;\;\;-3 \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-3, \left(x.im \cdot x.im\right) \cdot x.re\_m, {x.re\_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\frac{x.re\_m}{x.im}}{x.im}, x.re\_m, -3\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re\_m\\ \end{array} \end{array} \end{array} \]

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.im)) x.re_m)
          (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))))
   (*
    x.re_s
    (if (<= t_0 -2e+208)
      (* -3.0 (* (* x.im x.re_m) x.im))
      (if (<= t_0 INFINITY)
        (fma -3.0 (* (* x.im x.im) x.re_m) (pow x.re_m 3.0))
        (*
         (* (fma (/ (/ x.re_m x.im) x.im) x.re_m -3.0) (* x.im x.im))
         x.re_m))))))

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_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im);
	double tmp;
	if (t_0 <= -2e+208) {
		tmp = -3.0 * ((x_46_im * x_46_re_m) * x_46_im);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(-3.0, ((x_46_im * x_46_im) * x_46_re_m), pow(x_46_re_m, 3.0));
	} else {
		tmp = (fma(((x_46_re_m / x_46_im) / x_46_im), x_46_re_m, -3.0) * (x_46_im * x_46_im)) * x_46_re_m;
	}
	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(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im))
	tmp = 0.0
	if (t_0 <= -2e+208)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im * x_46_re_m) * x_46_im));
	elseif (t_0 <= Inf)
		tmp = fma(-3.0, Float64(Float64(x_46_im * x_46_im) * x_46_re_m), (x_46_re_m ^ 3.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_46_re_m / x_46_im) / x_46_im), x_46_re_m, -3.0) * Float64(x_46_im * x_46_im)) * x_46_re_m);
	end
	return Float64(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[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -2e+208], N[(-3.0 * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(-3.0 * N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x$46$re$95$m / x$46$im), $MachinePrecision] / x$46$im), $MachinePrecision] * x$46$re$95$m + -3.0), $MachinePrecision] * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2e208

   1. Initial program 85.6%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 39.8

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

   5. Applied rewrites 39.8%

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

   7. Applied rewrites 54.2%

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

   if -2e208 < (-.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)) < +inf.0

   1. Initial program 94.1%

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

   3. Taylor expanded in x.re around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-out N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

   5. Applied rewrites 92.4%

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

   if +inf.0 < (-.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 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. pow-prod-up N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out-- N/A

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

      12. +-commutative N/A

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

      13. associate--l- N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   7. Applied rewrites 42.3%

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

   8. Taylor expanded in x.im around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{x.re}{x.im}}{x.im}, x.re, -3\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+182}:\\ \;\;\;\;-3 \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\frac{x.re\_m}{x.im}}{x.im}, x.re\_m, -3\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re\_m\\ \end{array} \end{array} \end{array} \]

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.im)) x.re_m)
          (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))))
   (*
    x.re_s
    (if (<= t_0 -5e+182)
      (* -3.0 (* (* x.im x.re_m) x.im))
      (if (<= t_0 INFINITY)
        (* (fma -3.0 (* x.im x.im) (* x.re_m x.re_m)) x.re_m)
        (*
         (* (fma (/ (/ x.re_m x.im) x.im) x.re_m -3.0) (* x.im x.im))
         x.re_m))))))

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_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im);
	double tmp;
	if (t_0 <= -5e+182) {
		tmp = -3.0 * ((x_46_im * x_46_re_m) * x_46_im);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(-3.0, (x_46_im * x_46_im), (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = (fma(((x_46_re_m / x_46_im) / x_46_im), x_46_re_m, -3.0) * (x_46_im * x_46_im)) * x_46_re_m;
	}
	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(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im))
	tmp = 0.0
	if (t_0 <= -5e+182)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im * x_46_re_m) * x_46_im));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(-3.0, Float64(x_46_im * x_46_im), Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_46_re_m / x_46_im) / x_46_im), x_46_re_m, -3.0) * Float64(x_46_im * x_46_im)) * x_46_re_m);
	end
	return Float64(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[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -5e+182], N[(-3.0 * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x$46$re$95$m / x$46$im), $MachinePrecision] / x$46$im), $MachinePrecision] * x$46$re$95$m + -3.0), $MachinePrecision] * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.99999999999999973e182

   1. Initial program 86.0%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   7. Applied rewrites 55.5%

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

   if -4.99999999999999973e182 < (-.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)) < +inf.0

   1. Initial program 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. pow-prod-up N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out-- N/A

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

      12. +-commutative N/A

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

      13. associate--l- N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   7. Applied rewrites 94.1%

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

   if +inf.0 < (-.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 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. pow-prod-up N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out-- N/A

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

      12. +-commutative N/A

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

      13. associate--l- N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   7. Applied rewrites 42.3%

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

   8. Taylor expanded in x.im around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{x.re}{x.im}}{x.im}, x.re, -3\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -1 \cdot 10^{-291}:\\ \;\;\;\;-3 \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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 x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       -1e-291)
    (* -3.0 (* (* x.im x.re_m) x.im))
    (* (* x.re_m x.re_m) x.re_m))))

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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-291) {
		tmp = -3.0 * ((x_46_im * x_46_re_m) * x_46_im);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	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 * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46re_m * x_46im) + (x_46im * x_46re_m)) * x_46im)) <= (-1d-291)) then
        tmp = (-3.0d0) * ((x_46im * x_46re_m) * x_46im)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-291) {
		tmp = -3.0 * ((x_46_im * x_46_re_m) * x_46_im);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-291:
		tmp = -3.0 * ((x_46_im * x_46_re_m) * x_46_im)
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	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 (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -1e-291)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im * x_46_re_m) * x_46_im));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-291)
		tmp = -3.0 * ((x_46_im * x_46_re_m) * x_46_im);
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	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[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -1e-291], N[(-3.0 * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $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)) < -9.99999999999999962e-292

   1. Initial program 91.3%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   7. Applied rewrites 59.2%

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

   if -9.99999999999999962e-292 < (-.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 75.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. pow-prod-up N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 35.4%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out-- N/A

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

      12. +-commutative N/A

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

      13. associate--l- N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in x.re around inf

      \[\leadsto {x.re}^{2} \cdot x.re \]
   9. Step-by-step derivation

   10. Applied rewrites 69.6%

       \[\leadsto \left(x.re \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -1 \cdot 10^{-291}:\\ \;\;\;\;-3 \cdot \left(\left(x.im \cdot x.im\right) \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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 x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       -1e-291)
    (* -3.0 (* (* x.im x.im) x.re_m))
    (* (* x.re_m x.re_m) x.re_m))))

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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-291) {
		tmp = -3.0 * ((x_46_im * x_46_im) * x_46_re_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	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 * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46re_m * x_46im) + (x_46im * x_46re_m)) * x_46im)) <= (-1d-291)) then
        tmp = (-3.0d0) * ((x_46im * x_46im) * x_46re_m)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-291) {
		tmp = -3.0 * ((x_46_im * x_46_im) * x_46_re_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-291:
		tmp = -3.0 * ((x_46_im * x_46_im) * x_46_re_m)
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	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 (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -1e-291)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im * x_46_im) * x_46_re_m));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-291)
		tmp = -3.0 * ((x_46_im * x_46_im) * x_46_re_m);
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	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[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -1e-291], N[(-3.0 * N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $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)) < -9.99999999999999962e-292

   1. Initial program 91.3%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   if -9.99999999999999962e-292 < (-.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 75.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. pow-prod-up N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 35.4%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out-- N/A

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

      12. +-commutative N/A

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

      13. associate--l- N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in x.re around inf

      \[\leadsto {x.re}^{2} \cdot x.re \]
   9. Step-by-step derivation

   10. Applied rewrites 69.6%

       \[\leadsto \left(x.re \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 58.9% accurate, 3.6× speedup

Math

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

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.re_m)))

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_re_m);
}

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_46re_m)
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_re_m);
}

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_re_m)

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_re_m) * x_46_re_m))
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_re_m);
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$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval N/A

       \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   14. pow2 N/A

       \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   16. pow-prod-down N/A

       \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   17. pow-prod-up N/A

       \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   18. lower-pow.f64 N/A

       \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   19. metadata-eval N/A

       \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

4. Applied rewrites 33.0%

   \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

5. Taylor expanded in x.im around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. distribute-rgt-in N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. +-commutative N/A

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

   11. distribute-lft-out-- N/A

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

   12. +-commutative N/A

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

   13. associate--l- N/A

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

   14. sub-neg N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

7. Applied rewrites 86.7%

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

8. Taylor expanded in x.re around inf

   \[\leadsto {x.re}^{2} \cdot x.re \]
9. Step-by-step derivation

10. Applied rewrites 58.1%

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

Alternative 6: 23.9% accurate, 3.6× speedup

Math

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

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.im x.im) x.re_m)))

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_im * x_46_im) * x_46_re_m);
}

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_46im * x_46im) * x_46re_m)
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_im * x_46_im) * x_46_re_m);
}

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_im * x_46_im) * x_46_re_m)

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_im * x_46_im) * x_46_re_m))
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_im * x_46_im) * x_46_re_m);
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$46$im * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-inverses N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-neg-frac2 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

4. Applied rewrites 55.0%

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

5. Taylor expanded in x.re around 0

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-rgt-identity N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 23.1

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

7. Applied rewrites 23.1%

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

Developer Target 1: 87.3% 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 2024308 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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