math.cube on complex, real part

Percentage Accurate: 83.3% → 99.3%

Time: 12.3s

Alternatives: 9

Speedup: 0.7×

• 43.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.3% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+178}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot -3\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-3, \left(x.im \cdot x.im\right) \cdot x.re\_m, {x.re\_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, x.im + x.re\_m, 2 \cdot x.im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* (- (* x.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 -1e+178)
      (* x.im (* (* x.im x.re_m) -3.0))
      (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) (* 2.0 x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (((x_46_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 <= -1e+178) {
		tmp = x_46_im * ((x_46_im * x_46_re_m) * -3.0);
	} 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), (2.0 * x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(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 <= -1e+178)
		tmp = Float64(x_46_im * Float64(Float64(x_46_im * x_46_re_m) * -3.0));
	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 = fma(Float64(x_46_re_m - x_46_im), Float64(x_46_im + x_46_re_m), Float64(2.0 * x_46_im));
	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, -1e+178], N[(x$46$im * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * -3.0), $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[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re$95$m), $MachinePrecision] + N[(2.0 * x$46$im), $MachinePrecision]), $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)) < -1.0000000000000001e178

   1. Initial program 87.9%

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

   3. 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 30.3

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

   5. Applied rewrites 30.3%

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

   7. Applied rewrites 42.3%

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

   if -1.0000000000000001e178 < (-.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 96.5%

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

   3. Taylor expanded in x.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 95.9%

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

   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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 40.0%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+240}:\\ \;\;\;\;-3 \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, x.im + x.re\_m, 2 \cdot x.im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* (- (* x.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 -1e+240)
      (* -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) (* 2.0 x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (((x_46_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 <= -1e+240) {
		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), (2.0 * x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(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 <= -1e+240)
		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 = fma(Float64(x_46_re_m - x_46_im), Float64(x_46_im + x_46_re_m), Float64(2.0 * x_46_im));
	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, -1e+240], 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[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re$95$m), $MachinePrecision] + N[(2.0 * x$46$im), $MachinePrecision]), $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)) < -1.00000000000000001e240

   1. Initial program 87.2%

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

   3. Taylor expanded in x.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 32.0

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

   5. Applied rewrites 32.0%

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

   7. Applied rewrites 44.7%

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

   if -1.00000000000000001e240 < (-.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 96.5%

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

      1. 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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{{x.re}^{\color{blue}{6}} - {\left(x.im \cdot x.im\right)}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      11. pow2 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. pow-pow N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. metadata-eval N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{\color{blue}{6}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\left(x.im \cdot x.im\right) \cdot \color{blue}{\left(x.re \cdot x.re + x.im \cdot x.im\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\mathsf{fma}\left(x.im \cdot x.im, x.re \cdot x.re + x.im \cdot x.im, \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 32.9%

      \[\leadsto \color{blue}{\frac{{x.re}^{6} - {x.im}^{6}}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. unpow3 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 96.5%

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

   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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 40.0%

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

Alternative 3: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-323}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot -3\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, x.im + x.re\_m, 2 \cdot x.im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* (- (* x.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-323)
      (* x.im (* (* x.im x.re_m) -3.0))
      (if (<= t_0 INFINITY)
        (* (* x.re_m x.re_m) x.re_m)
        (fma (- x.re_m x.im) (+ x.im x.re_m) (* 2.0 x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (((x_46_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-323) {
		tmp = x_46_im * ((x_46_im * x_46_re_m) * -3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (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), (2.0 * x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(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-323)
		tmp = Float64(x_46_im * Float64(Float64(x_46_im * x_46_re_m) * -3.0));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(x_46_im + x_46_re_m), Float64(2.0 * x_46_im));
	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-323], N[(x$46$im * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re$95$m), $MachinePrecision] + N[(2.0 * x$46$im), $MachinePrecision]), $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)) < -1.97626e-323

   1. Initial program 92.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 40.7

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

   5. Applied rewrites 40.7%

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

   7. Applied rewrites 47.9%

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

   if -1.97626e-323 < (-.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 95.6%

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

      1. 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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{{x.re}^{\color{blue}{6}} - {\left(x.im \cdot x.im\right)}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      11. pow2 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. pow-pow N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. metadata-eval N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{\color{blue}{6}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\left(x.im \cdot x.im\right) \cdot \color{blue}{\left(x.re \cdot x.re + x.im \cdot x.im\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\mathsf{fma}\left(x.im \cdot x.im, x.re \cdot x.re + x.im \cdot x.im, \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\frac{{x.re}^{6} - {x.im}^{6}}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. unpow3 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 95.6%

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

   8. Taylor expanded in x.re around inf

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

   10. Applied rewrites 65.1%

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

   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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 40.0%

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

Alternative 4: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -2 \cdot 10^{-323}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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))
       -2e-323)
    (* x.im (* (* x.im x.re_m) -3.0))
    (* (* 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)) <= -2e-323) {
		tmp = x_46_im * ((x_46_im * x_46_re_m) * -3.0);
	} 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)) <= (-2d-323)) then
        tmp = x_46im * ((x_46im * x_46re_m) * (-3.0d0))
    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)) <= -2e-323) {
		tmp = x_46_im * ((x_46_im * x_46_re_m) * -3.0);
	} 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)) <= -2e-323:
		tmp = x_46_im * ((x_46_im * x_46_re_m) * -3.0)
	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)) <= -2e-323)
		tmp = Float64(x_46_im * Float64(Float64(x_46_im * x_46_re_m) * -3.0));
	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)) <= -2e-323)
		tmp = x_46_im * ((x_46_im * x_46_re_m) * -3.0);
	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], -2e-323], N[(x$46$im * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * -3.0), $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)) < -1.97626e-323

   1. Initial program 92.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 40.7

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

   5. Applied rewrites 40.7%

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

   7. Applied rewrites 47.9%

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

   if -1.97626e-323 < (-.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 78.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{{x.re}^{\color{blue}{6}} - {\left(x.im \cdot x.im\right)}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      11. pow2 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. pow-pow N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. metadata-eval N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{\color{blue}{6}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\left(x.im \cdot x.im\right) \cdot \color{blue}{\left(x.re \cdot x.re + x.im \cdot x.im\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\mathsf{fma}\left(x.im \cdot x.im, x.re \cdot x.re + x.im \cdot x.im, \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 20.2%

      \[\leadsto \color{blue}{\frac{{x.re}^{6} - {x.im}^{6}}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. unpow3 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 87.2%

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

   8. Taylor expanded in x.re around inf

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

   10. Applied rewrites 68.4%

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

Alternative 5: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -2 \cdot 10^{-323}:\\ \;\;\;\;-3 \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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))
       -2e-323)
    (* -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)) <= -2e-323) {
		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)) <= (-2d-323)) 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)) <= -2e-323) {
		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)) <= -2e-323:
		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)) <= -2e-323)
		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)) <= -2e-323)
		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], -2e-323], 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)) < -1.97626e-323

   1. Initial program 92.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 40.7

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

   5. Applied rewrites 40.7%

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

   7. Applied rewrites 47.9%

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

   if -1.97626e-323 < (-.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 78.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{{x.re}^{\color{blue}{6}} - {\left(x.im \cdot x.im\right)}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      11. pow2 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. pow-pow N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. metadata-eval N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{\color{blue}{6}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\left(x.im \cdot x.im\right) \cdot \color{blue}{\left(x.re \cdot x.re + x.im \cdot x.im\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\mathsf{fma}\left(x.im \cdot x.im, x.re \cdot x.re + x.im \cdot x.im, \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 20.2%

      \[\leadsto \color{blue}{\frac{{x.re}^{6} - {x.im}^{6}}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. unpow3 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 87.2%

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

   8. Taylor expanded in x.re around inf

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

   10. Applied rewrites 68.4%

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

Alternative 6: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -2 \cdot 10^{-323}:\\ \;\;\;\;-3 \cdot \left(\left(x.im \cdot x.im\right) \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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))
       -2e-323)
    (* -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)) <= -2e-323) {
		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)) <= (-2d-323)) 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)) <= -2e-323) {
		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)) <= -2e-323:
		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)) <= -2e-323)
		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)) <= -2e-323)
		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], -2e-323], 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)) < -1.97626e-323

   1. Initial program 92.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 40.7

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

   5. Applied rewrites 40.7%

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

   if -1.97626e-323 < (-.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 78.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{{x.re}^{\color{blue}{6}} - {\left(x.im \cdot x.im\right)}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      11. pow2 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. pow-pow N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. metadata-eval N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{\color{blue}{6}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\left(x.im \cdot x.im\right) \cdot \color{blue}{\left(x.re \cdot x.re + x.im \cdot x.im\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\mathsf{fma}\left(x.im \cdot x.im, x.re \cdot x.re + x.im \cdot x.im, \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 20.2%

      \[\leadsto \color{blue}{\frac{{x.re}^{6} - {x.im}^{6}}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. unpow3 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 87.2%

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

   8. Taylor expanded in x.re around inf

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

   10. Applied rewrites 68.4%

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

Alternative 7: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -2 \cdot 10^{-323}:\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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))
       -2e-323)
    (* (- 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)) <= -2e-323) {
		tmp = -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)) <= (-2d-323)) then
        tmp = -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)) <= -2e-323) {
		tmp = -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)) <= -2e-323:
		tmp = -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)) <= -2e-323)
		tmp = Float64(Float64(-x_46_im) * Float64(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)) <= -2e-323)
		tmp = -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], -2e-323], N[((-x$46$im) * N[(x$46$im * 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)) < -1.97626e-323

   1. Initial program 92.6%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 40.7

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

   5. Applied rewrites 40.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. flip-+ N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. lower-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   7. Applied rewrites 19.3%

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

   8. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 23.6

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

   10. Applied rewrites 23.6%

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

   if -1.97626e-323 < (-.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 78.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{{x.re}^{\color{blue}{6}} - {\left(x.im \cdot x.im\right)}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      11. pow2 N/A

          \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. pow-pow N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. metadata-eval N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{\color{blue}{6}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. +-commutative N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\left(x.im \cdot x.im\right) \cdot \color{blue}{\left(x.re \cdot x.re + x.im \cdot x.im\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\mathsf{fma}\left(x.im \cdot x.im, x.re \cdot x.re + x.im \cdot x.im, \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 20.2%

      \[\leadsto \color{blue}{\frac{{x.re}^{6} - {x.im}^{6}}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. unpow3 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 87.2%

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

   8. Taylor expanded in x.re around inf

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

   10. Applied rewrites 68.4%

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

Alternative 8: 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 83.3%

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

   1. 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. flip3-- N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. pow-pow N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

      \[\leadsto \frac{{x.re}^{\color{blue}{6}} - {\left(x.im \cdot x.im\right)}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   11. pow2 N/A

       \[\leadsto \frac{{x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   12. pow-pow N/A

       \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{{x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   14. metadata-eval N/A

       \[\leadsto \frac{{x.re}^{6} - {x.im}^{\color{blue}{6}}}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   15. +-commutative N/A

       \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   16. distribute-rgt-out N/A

       \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   17. +-commutative N/A

       \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\left(x.im \cdot x.im\right) \cdot \color{blue}{\left(x.re \cdot x.re + x.im \cdot x.im\right)} + \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   18. lower-fma.f64 N/A

       \[\leadsto \frac{{x.re}^{6} - {x.im}^{6}}{\color{blue}{\mathsf{fma}\left(x.im \cdot x.im, x.re \cdot x.re + x.im \cdot x.im, \left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

4. Applied rewrites 22.3%

   \[\leadsto \color{blue}{\frac{{x.re}^{6} - {x.im}^{6}}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

5. Taylor expanded in x.im around 0

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

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

   5. distribute-rgt1-in N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. cancel-sign-sub-inv N/A

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

   10. metadata-eval N/A

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

   11. unpow3 N/A

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. distribute-rgt-in N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

7. Applied rewrites 89.2%

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

8. Taylor expanded in x.re around inf

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

10. Applied rewrites 63.2%

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

Alternative 9: 3.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(2 \cdot x.im\right) \end{array} \]

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 (* 2.0 x.im)))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (2.0 * x_46_im);
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * (2.0d0 * x_46im)
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (2.0 * x_46_im);
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * (2.0 * x_46_im)

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(2.0 * x_46_im))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * (2.0 * x_46_im);
end

Wolfram

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

Derivation

1. Initial program 83.3%

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

3. Taylor expanded in x.re around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 49.0

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

5. Applied rewrites 49.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. flip-+ N/A

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

   10. +-inverses N/A

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

   11. +-inverses N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. +-inverses N/A

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

   15. +-inverses N/A

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

   16. flip-+ N/A

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

   17. lower-+.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f64 N/A

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

7. Applied rewrites 16.6%

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

8. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{2 \cdot x.im} \]
9. Step-by-step derivation

   1. lower-*.f64 3.2

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

10. Applied rewrites 3.2%

    \[\leadsto \color{blue}{2 \cdot x.im} \]
11. Add Preprocessing

Developer Target 1: 87.8% 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 2024313 
(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)))