math.cube on complex, imaginary part

Percentage Accurate: 82.8% → 99.7%

Time: 6.1s

Alternatives: 8

Speedup: 0.9×

• 45.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.8% accurate, 1.0× speedup

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := t\_0 + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{+281}:\\ \;\;\;\;t\_0 + \left(x.re \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.re\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.re + x.im\_m, x.re - x.im\_m, 2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))
        (t_1 (+ t_0 (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_1 1e+281)
      (+ t_0 (* (* x.re (+ x.im_m x.im_m)) x.re))
      (if (<= t_1 INFINITY)
        (* 3.0 (* (* x.re x.im_m) x.re))
        (* x.im_m (fma (+ x.re x.im_m) (- x.re x.im_m) 2.0)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m;
	double t_1 = t_0 + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= 1e+281) {
		tmp = t_0 + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 3.0 * ((x_46_re * x_46_im_m) * x_46_re);
	} else {
		tmp = x_46_im_m * fma((x_46_re + x_46_im_m), (x_46_re - x_46_im_m), 2.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m)
	t_1 = Float64(t_0 + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_1 <= 1e+281)
		tmp = Float64(t_0 + Float64(Float64(x_46_re * Float64(x_46_im_m + x_46_im_m)) * x_46_re));
	elseif (t_1 <= Inf)
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_im_m) * x_46_re));
	else
		tmp = Float64(x_46_im_m * fma(Float64(x_46_re + x_46_im_m), Float64(x_46_re - x_46_im_m), 2.0));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, 1e+281], N[(t$95$0 + N[(N[(x$46$re * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(3.0 * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision] + 2.0), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 1e281

   1. Initial program 97.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 97.1

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

   4. Applied rewrites 97.1%

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

   if 1e281 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 84.2%

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

   3. Taylor expanded in x.re around inf

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 63.2%

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

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 25.9

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

   4. Applied rewrites 25.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 25.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 25.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-in N/A

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

      22. lift-+.f64 N/A

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

      23. *-commutative N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x.re around 0

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

   9. Applied rewrites 100.0%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re \cdot x.re\right) \cdot 3, x.im\_m, \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.re + x.im\_m, x.re - x.im\_m, 2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_0 2e+65)
      (fma (* (* x.re x.re) 3.0) x.im_m (* (* (- x.im_m) x.im_m) x.im_m))
      (if (<= t_0 INFINITY)
        (* 3.0 (* (* x.re x.im_m) x.re))
        (* x.im_m (fma (+ x.re x.im_m) (- x.re x.im_m) 2.0)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= 2e+65) {
		tmp = fma(((x_46_re * x_46_re) * 3.0), x_46_im_m, ((-x_46_im_m * x_46_im_m) * x_46_im_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 3.0 * ((x_46_re * x_46_im_m) * x_46_re);
	} else {
		tmp = x_46_im_m * fma((x_46_re + x_46_im_m), (x_46_re - x_46_im_m), 2.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= 2e+65)
		tmp = fma(Float64(Float64(x_46_re * x_46_re) * 3.0), x_46_im_m, Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m));
	elseif (t_0 <= Inf)
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_im_m) * x_46_re));
	else
		tmp = Float64(x_46_im_m * fma(Float64(x_46_re + x_46_im_m), Float64(x_46_re - x_46_im_m), 2.0));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 2e+65], N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$im$95$m + N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(3.0 * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision] + 2.0), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 2e65

   1. Initial program 96.8%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 96.7%

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

   7. Applied rewrites 93.5%

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

   if 2e65 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 88.4%

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

   3. Taylor expanded in x.re around inf

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

   5. Applied rewrites 59.6%

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

   7. Applied rewrites 59.7%

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

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 25.9

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

   4. Applied rewrites 25.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 25.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 25.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-in N/A

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

      22. lift-+.f64 N/A

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

      23. *-commutative N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x.re around 0

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

   9. Applied rewrites 100.0%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(3 \cdot x.re, x.re, \left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.re + x.im\_m, x.re - x.im\_m, 2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_0 2e+65)
      (* (fma (* 3.0 x.re) x.re (* (- x.im_m) x.im_m)) x.im_m)
      (if (<= t_0 INFINITY)
        (* 3.0 (* (* x.re x.im_m) x.re))
        (* x.im_m (fma (+ x.re x.im_m) (- x.re x.im_m) 2.0)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= 2e+65) {
		tmp = fma((3.0 * x_46_re), x_46_re, (-x_46_im_m * x_46_im_m)) * x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 3.0 * ((x_46_re * x_46_im_m) * x_46_re);
	} else {
		tmp = x_46_im_m * fma((x_46_re + x_46_im_m), (x_46_re - x_46_im_m), 2.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= 2e+65)
		tmp = Float64(fma(Float64(3.0 * x_46_re), x_46_re, Float64(Float64(-x_46_im_m) * x_46_im_m)) * x_46_im_m);
	elseif (t_0 <= Inf)
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_im_m) * x_46_re));
	else
		tmp = Float64(x_46_im_m * fma(Float64(x_46_re + x_46_im_m), Float64(x_46_re - x_46_im_m), 2.0));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 2e+65], N[(N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$re + N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(3.0 * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision] + 2.0), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 2e65

   1. Initial program 96.8%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 96.7%

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

   if 2e65 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 88.4%

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

   3. Taylor expanded in x.re around inf

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

   5. Applied rewrites 59.6%

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

   7. Applied rewrites 59.7%

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

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 25.9

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

   4. Applied rewrites 25.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 25.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 25.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-in N/A

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

      22. lift-+.f64 N/A

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

      23. *-commutative N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x.re around 0

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

   9. Applied rewrites 100.0%

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

Alternative 4: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.re + x.im\_m, x.re - x.im\_m, 2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_0 -5e-292)
      (* (* (- x.im_m) x.im_m) x.im_m)
      (if (<= t_0 INFINITY)
        (* 3.0 (* (* x.re x.im_m) x.re))
        (* x.im_m (fma (+ x.re x.im_m) (- x.re x.im_m) 2.0)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= -5e-292) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 3.0 * ((x_46_re * x_46_im_m) * x_46_re);
	} else {
		tmp = x_46_im_m * fma((x_46_re + x_46_im_m), (x_46_re - x_46_im_m), 2.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= -5e-292)
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	elseif (t_0 <= Inf)
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_im_m) * x_46_re));
	else
		tmp = Float64(x_46_im_m * fma(Float64(x_46_re + x_46_im_m), Float64(x_46_re - x_46_im_m), 2.0));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -5e-292], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(3.0 * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision] + 2.0), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.99999999999999981e-292

   1. Initial program 95.2%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 49.2%

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

   if -4.99999999999999981e-292 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 93.2%

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

   3. Taylor expanded in x.re around inf

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 72.8%

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

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 25.9

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

   4. Applied rewrites 25.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 25.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 25.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-in N/A

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

      22. lift-+.f64 N/A

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

      23. *-commutative N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x.re around 0

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

   9. Applied rewrites 100.0%

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

Alternative 5: 95.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-292} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot x.re\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -5e-292) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* 3.0 (* (* x.re x.im_m) x.re))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -5e-292) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = 3.0 * ((x_46_re * x_46_im_m) * x_46_re);
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -5e-292) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = 3.0 * ((x_46_re * x_46_im_m) * x_46_re);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -5e-292) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = 3.0 * ((x_46_re * x_46_im_m) * x_46_re)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -5e-292) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_im_m) * x_46_re));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -5e-292) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = 3.0 * ((x_46_re * x_46_im_m) * x_46_re);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -5e-292], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(3.0 * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.99999999999999981e-292 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 75.1%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 54.5%

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

   if -4.99999999999999981e-292 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 93.2%

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

   3. Taylor expanded in x.re around inf

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 72.8%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-292} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.im\right) \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-292} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -5e-292) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* (* x.im_m x.re) x.re)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -5e-292) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_im_m * x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -5e-292) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_im_m * x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -5e-292) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = (x_46_im_m * x_46_re) * x_46_re
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -5e-292) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_im_m * x_46_re) * x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -5e-292) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = (x_46_im_m * x_46_re) * x_46_re;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -5e-292], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * x$46$re), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.99999999999999981e-292 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 75.1%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 54.5%

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

   if -4.99999999999999981e-292 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 93.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-in N/A

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

      22. lift-+.f64 N/A

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

      23. *-commutative N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x.re around inf

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

   9. Applied rewrites 43.4%

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

   11. Applied rewrites 44.2%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-292} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4 \cdot 10^{+50}:\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(\left(x.re - x.im\_m\right) \cdot x.im\_m\right) + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.re + x.im\_m, x.re - x.im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 4e+50)
    (+
     (* (+ x.im_m x.re) (* (- x.re x.im_m) x.im_m))
     (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))
    (* x.im_m (fma (+ x.re x.im_m) (- x.re x.im_m) 2.0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 4e+50) {
		tmp = ((x_46_im_m + x_46_re) * ((x_46_re - x_46_im_m) * x_46_im_m)) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	} else {
		tmp = x_46_im_m * fma((x_46_re + x_46_im_m), (x_46_re - x_46_im_m), 2.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 4e+50)
		tmp = Float64(Float64(Float64(x_46_im_m + x_46_re) * Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m)) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re));
	else
		tmp = Float64(x_46_im_m * fma(Float64(x_46_re + x_46_im_m), Float64(x_46_re - x_46_im_m), 2.0));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4e+50], N[(N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.0000000000000003e50

   1. Initial program 87.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 96.2

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

   4. Applied rewrites 96.2%

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

   if 4.0000000000000003e50 < x.im

   1. Initial program 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 77.9

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

   4. Applied rewrites 77.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 77.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 77.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-in N/A

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

      22. lift-+.f64 N/A

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

      23. *-commutative N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x.re around 0

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

   9. Applied rewrites 100.0%

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

Alternative 8: 35.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * ((x_46_im_m * x_46_re) * x_46_re);
}

Fortran

x.im\_m =     private
x.im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46im_s, x_46re, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * ((x_46im_m * x_46re) * x_46re)
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * ((x_46_im_m * x_46_re) * x_46_re);
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	return x_46_im_s * ((x_46_im_m * x_46_re) * x_46_re)

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(x_46_im_m * x_46_re) * x_46_re))
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * ((x_46_im_m * x_46_re) * x_46_re);
end

Wolfram

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

Derivation

1. Initial program 84.2%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. difference-of-squares N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 92.0

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

4. Applied rewrites 92.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. lower-fma.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lift-+.f64 N/A

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

   11. lower-*.f64 92.0

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 92.0

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. distribute-lft-in N/A

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

   22. lift-+.f64 N/A

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

   23. *-commutative N/A

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

6. Applied rewrites 60.0%

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

7. Taylor expanded in x.re around inf

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

9. Applied rewrites 33.2%

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

11. Applied rewrites 33.9%

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

Developer Target 1: 91.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - 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_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025019 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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