math.cube on complex, real part

Percentage Accurate: 83.3% → 99.7%

Time: 8.3s

Alternatives: 6

Speedup: 1.4×

• 44.0% 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: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 3.7 \cdot 10^{+101}:\\ \;\;\;\;\left(x.re\_m \cdot \left(x.im\_m + x.re\_m\right)\right) \cdot \left(x.re\_m - x.im\_m\right) - \left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, \mathsf{fma}\left(\frac{x.re\_m}{x.im\_m}, x.re\_m, x.re\_m\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 3.7e+101)
    (-
     (* (* x.re_m (+ x.im_m x.re_m)) (- x.re_m x.im_m))
     (* (* x.re_m (+ x.im_m x.im_m)) x.im_m))
    (fma
     (- x.re_m x.im_m)
     (* (fma (/ x.re_m x.im_m) x.re_m x.re_m) x.im_m)
     (* 2.0 x.im_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 3.7e+101) {
		tmp = ((x_46_re_m * (x_46_im_m + x_46_re_m)) * (x_46_re_m - x_46_im_m)) - ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_im_m);
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (fma((x_46_re_m / x_46_im_m), x_46_re_m, x_46_re_m) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 3.7e+101)
		tmp = Float64(Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_re_m)) * Float64(x_46_re_m - x_46_im_m)) - Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)) * x_46_im_m));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(fma(Float64(x_46_re_m / x_46_im_m), x_46_re_m, x_46_re_m) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 3.7e+101], N[(N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(N[(N[(x$46$re$95$m / x$46$im$95$m), $MachinePrecision] * x$46$re$95$m + x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.6999999999999997e101

   1. Initial program 88.5%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 95.6

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

   4. Applied rewrites 95.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 95.6

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

   6. Applied rewrites 95.6%

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

   if 3.6999999999999997e101 < x.re

   1. Initial program 68.3%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 80.5

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

   4. Applied rewrites 80.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 80.5

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

   6. Applied rewrites 80.5%

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

   7. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 80.5

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

   9. Applied rewrites 80.5%

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

   11. Applied rewrites 100.0%

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

Alternative 2: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -5 \cdot 10^{-70}:\\ \;\;\;\;-3 \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m, \left(x.im\_m \cdot x.im\_m\right) \cdot -3\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       -5e-70)
    (* -3.0 (* (* x.im_m x.re_m) x.im_m))
    (* (fma x.re_m x.re_m (* (* x.im_m x.im_m) -3.0)) x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-70) {
		tmp = -3.0 * ((x_46_im_m * x_46_re_m) * x_46_im_m);
	} else {
		tmp = fma(x_46_re_m, x_46_re_m, ((x_46_im_m * x_46_im_m) * -3.0)) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-70)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im_m * x_46_re_m) * x_46_im_m));
	else
		tmp = Float64(fma(x_46_re_m, x_46_re_m, Float64(Float64(x_46_im_m * x_46_im_m) * -3.0)) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -5e-70], N[(-3.0 * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m + N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $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)) < -4.9999999999999998e-70

   1. Initial program 93.8%

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

   3. Taylor expanded in x.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. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   if -4.9999999999999998e-70 < (-.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 80.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. pow-prod-up N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow-prod-down N/A

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

      14. pow-prod-up N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 42.3

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

   4. Applied rewrites 42.3%

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

   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. mul-1-neg N/A

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow3 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 87.7

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

   7. Applied rewrites 87.7%

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

   9. Applied rewrites 94.4%

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

Alternative 3: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -2 \cdot 10^{-321}:\\ \;\;\;\;-3 \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       -2e-321)
    (* -3.0 (* (* x.im_m x.re_m) x.im_m))
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321) {
		tmp = -3.0 * ((x_46_im_m * x_46_re_m) * x_46_im_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46re_m * x_46im_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-2d-321)) then
        tmp = (-3.0d0) * ((x_46im_m * x_46re_m) * x_46im_m)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321) {
		tmp = -3.0 * ((x_46_im_m * x_46_re_m) * x_46_im_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321:
		tmp = -3.0 * ((x_46_im_m * x_46_re_m) * x_46_im_m)
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im_m * x_46_re_m) * x_46_im_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.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321)
		tmp = -3.0 * ((x_46_im_m * x_46_re_m) * x_46_im_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.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -2e-321], N[(-3.0 * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im$95$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)) < -2.00097e-321

   1. Initial program 94.7%

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

   3. 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. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if -2.00097e-321 < (-.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 78.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. pow-prod-up N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow-prod-down N/A

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

      14. pow-prod-up N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 37.8

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

   4. Applied rewrites 37.8%

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

   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. mul-1-neg N/A

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow3 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 86.2

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

   7. Applied rewrites 86.2%

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

   9. Applied rewrites 93.8%

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

   10. Taylor expanded in x.re around inf

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

   12. Applied rewrites 64.8%

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

Alternative 4: 91.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -2 \cdot 10^{-321}:\\ \;\;\;\;-3 \cdot \left(\left(x.im\_m \cdot x.im\_m\right) \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       -2e-321)
    (* -3.0 (* (* x.im_m x.im_m) x.re_m))
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321) {
		tmp = -3.0 * ((x_46_im_m * x_46_im_m) * 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.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46re_m * x_46im_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-2d-321)) then
        tmp = (-3.0d0) * ((x_46im_m * x_46im_m) * 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.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321) {
		tmp = -3.0 * ((x_46_im_m * x_46_im_m) * 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.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321:
		tmp = -3.0 * ((x_46_im_m * x_46_im_m) * 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.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im_m * x_46_im_m) * 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.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321)
		tmp = -3.0 * ((x_46_im_m * x_46_im_m) * 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.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -2e-321], N[(-3.0 * N[(N[(x$46$im$95$m * x$46$im$95$m), $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)) < -2.00097e-321

   1. Initial program 94.7%

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

   3. 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. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 51.4%

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

   if -2.00097e-321 < (-.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 78.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. pow-prod-up N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow-prod-down N/A

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

      14. pow-prod-up N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 37.8

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

   4. Applied rewrites 37.8%

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

   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. mul-1-neg N/A

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow3 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 86.2

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

   7. Applied rewrites 86.2%

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

   9. Applied rewrites 93.8%

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

   10. Taylor expanded in x.re around inf

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

   12. Applied rewrites 64.8%

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

Alternative 5: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im\_m \cdot x.im\_m, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-3 \cdot x.re\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 2.4e+153)
    (* (fma -3.0 (* x.im_m x.im_m) (* x.re_m x.re_m)) x.re_m)
    (* (* (* -3.0 x.re_m) x.im_m) x.im_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.4e+153) {
		tmp = fma(-3.0, (x_46_im_m * x_46_im_m), (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2.4e+153)
		tmp = Float64(fma(-3.0, Float64(x_46_im_m * x_46_im_m), Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = Float64(Float64(Float64(-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 2.4e+153], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.39999999999999992e153

   1. Initial program 89.4%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if 2.39999999999999992e153 < x.im

   1. Initial program 60.7%

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

   3. 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. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.3%

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

Alternative 6: 58.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
2. Add Preprocessing
3. 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. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow-prod-down N/A

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

   8. pow-prod-up N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. pow-prod-down N/A

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

   14. pow-prod-up N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 38.6

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

4. Applied rewrites 38.6%

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

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. mul-1-neg N/A

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

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

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

   3. distribute-rgt1-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. unpow3 N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. distribute-rgt-in N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 89.9

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

7. Applied rewrites 89.9%

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

9. Applied rewrites 94.2%

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

10. Taylor expanded in x.re around inf

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

12. Applied rewrites 56.6%

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

Developer Target 1: 99.8% 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 2024337 
(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)))