math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 99.8%

Time: 1.8s

Alternatives: 8

Speedup: 0.9×

• 55.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 82.7% accurate, 1.0× speedup

Math

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

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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(x.re \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 + \left(x.re \cdot \left|x.im\right| + \left|x.im\right| \cdot x.re\right) \cdot x.re \leq 9999999999999999813486777206230041577815560719820581330098483720446847883279500839884297726782854580737362697004022581572770293687044935910015528960168049498887207223940204684198896264456339658487887951484580004902758521100414464490983962613190835886243290260424727924570510530141380583845003264:\\ \;\;\;\;t\_0 + \left(\left(x.re + x.re\right) \cdot \left|x.im\right|\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left|x.im\right| \cdot \left(\left(x.re + x.re\right) - \frac{\left|x.im\right| + x.re}{x.re} \cdot \left(\left|x.im\right| - x.re\right)\right)\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0
        (*
         (- (* x.re x.re) (* (fabs x.im) (fabs x.im)))
         (fabs x.im))))
  (*
   (copysign 1 x.im)
   (if (<=
        (+ t_0 (* (+ (* x.re (fabs x.im)) (* (fabs x.im) x.re)) x.re))
        9999999999999999813486777206230041577815560719820581330098483720446847883279500839884297726782854580737362697004022581572770293687044935910015528960168049498887207223940204684198896264456339658487887951484580004902758521100414464490983962613190835886243290260424727924570510530141380583845003264)
     (+ t_0 (* (* (+ x.re x.re) (fabs x.im)) x.re))
     (*
      (*
       (fabs x.im)
       (-
        (+ x.re x.re)
        (* (/ (+ (fabs x.im) x.re) x.re) (- (fabs x.im) x.re))))
      x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = ((x_46_re * x_46_re) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im);
	double tmp;
	if ((t_0 + (((x_46_re * fabs(x_46_im)) + (fabs(x_46_im) * x_46_re)) * x_46_re)) <= 1e+295) {
		tmp = t_0 + (((x_46_re + x_46_re) * fabs(x_46_im)) * x_46_re);
	} else {
		tmp = (fabs(x_46_im) * ((x_46_re + x_46_re) - (((fabs(x_46_im) + x_46_re) / x_46_re) * (fabs(x_46_im) - x_46_re)))) * x_46_re;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = ((x_46_re * x_46_re) - (Math.abs(x_46_im) * Math.abs(x_46_im))) * Math.abs(x_46_im);
	double tmp;
	if ((t_0 + (((x_46_re * Math.abs(x_46_im)) + (Math.abs(x_46_im) * x_46_re)) * x_46_re)) <= 1e+295) {
		tmp = t_0 + (((x_46_re + x_46_re) * Math.abs(x_46_im)) * x_46_re);
	} else {
		tmp = (Math.abs(x_46_im) * ((x_46_re + x_46_re) - (((Math.abs(x_46_im) + x_46_re) / x_46_re) * (Math.abs(x_46_im) - x_46_re)))) * x_46_re;
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = ((x_46_re * x_46_re) - (math.fabs(x_46_im) * math.fabs(x_46_im))) * math.fabs(x_46_im)
	tmp = 0
	if (t_0 + (((x_46_re * math.fabs(x_46_im)) + (math.fabs(x_46_im) * x_46_re)) * x_46_re)) <= 1e+295:
		tmp = t_0 + (((x_46_re + x_46_re) * math.fabs(x_46_im)) * x_46_re)
	else:
		tmp = (math.fabs(x_46_im) * ((x_46_re + x_46_re) - (((math.fabs(x_46_im) + x_46_re) / x_46_re) * (math.fabs(x_46_im) - x_46_re)))) * x_46_re
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + Float64(abs(x_46_im) * x_46_re)) * x_46_re)) <= 1e+295)
		tmp = Float64(t_0 + Float64(Float64(Float64(x_46_re + x_46_re) * abs(x_46_im)) * x_46_re));
	else
		tmp = Float64(Float64(abs(x_46_im) * Float64(Float64(x_46_re + x_46_re) - Float64(Float64(Float64(abs(x_46_im) + x_46_re) / x_46_re) * Float64(abs(x_46_im) - x_46_re)))) * x_46_re);
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = ((x_46_re * x_46_re) - (abs(x_46_im) * abs(x_46_im))) * abs(x_46_im);
	tmp = 0.0;
	if ((t_0 + (((x_46_re * abs(x_46_im)) + (abs(x_46_im) * x_46_re)) * x_46_re)) <= 1e+295)
		tmp = t_0 + (((x_46_re + x_46_re) * abs(x_46_im)) * x_46_re);
	else
		tmp = (abs(x_46_im) * ((x_46_re + x_46_re) - (((abs(x_46_im) + x_46_re) / x_46_re) * (abs(x_46_im) - x_46_re)))) * x_46_re;
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$0 + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], 9999999999999999813486777206230041577815560719820581330098483720446847883279500839884297726782854580737362697004022581572770293687044935910015528960168049498887207223940204684198896264456339658487887951484580004902758521100414464490983962613190835886243290260424727924570510530141380583845003264], N[(t$95$0 + N[(N[(N[(x$46$re + x$46$re), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(x$46$re + x$46$re), $MachinePrecision] - N[(N[(N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision] / x$46$re), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < 9.9999999999999998e294

   1. Initial program 82.7%

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

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 82.7%

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

   3. Applied rewrites 82.7%

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

   if 9.9999999999999998e294 < (+.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 82.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. sub-flip-reverse N/A

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

   3. Applied rewrites 91.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-+.f64 N/A

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

      6. lower-unsound-/.f64 89.8%

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

   5. Applied rewrites 89.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.5%

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. add-to-fraction N/A

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

      14. add-flip N/A

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

      15. *-lft-identity N/A

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

      16. add-flip N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lower-/.f64 91.5%

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

   7. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 95.9%

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

   9. Applied rewrites 95.9%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 1449999999999999918156969066552575526807194060320427185510591162486664329368057600973507395584:\\ \;\;\;\;3 \cdot \left(\left(\left|x.im\right| \cdot x.re\right) \cdot x.re\right) - \left(\left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \left(x.re \cdot \left(\left(x.re + x.re\right) - \frac{\left|x.im\right|}{x.re} \cdot \left(\left|x.im\right| - x.re\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1 x.im)
 (if (<=
      (fabs x.im)
      1449999999999999918156969066552575526807194060320427185510591162486664329368057600973507395584)
   (-
    (* 3 (* (* (fabs x.im) x.re) x.re))
    (* (* (fabs x.im) (fabs x.im)) (fabs x.im)))
   (*
    (fabs x.im)
    (*
     x.re
     (-
      (+ x.re x.re)
      (* (/ (fabs x.im) x.re) (- (fabs x.im) x.re))))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 1.45e+93) {
		tmp = (3.0 * ((fabs(x_46_im) * x_46_re) * x_46_re)) - ((fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im));
	} else {
		tmp = fabs(x_46_im) * (x_46_re * ((x_46_re + x_46_re) - ((fabs(x_46_im) / x_46_re) * (fabs(x_46_im) - x_46_re))));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (Math.abs(x_46_im) <= 1.45e+93) {
		tmp = (3.0 * ((Math.abs(x_46_im) * x_46_re) * x_46_re)) - ((Math.abs(x_46_im) * Math.abs(x_46_im)) * Math.abs(x_46_im));
	} else {
		tmp = Math.abs(x_46_im) * (x_46_re * ((x_46_re + x_46_re) - ((Math.abs(x_46_im) / x_46_re) * (Math.abs(x_46_im) - x_46_re))));
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_im) <= 1.45e+93:
		tmp = (3.0 * ((math.fabs(x_46_im) * x_46_re) * x_46_re)) - ((math.fabs(x_46_im) * math.fabs(x_46_im)) * math.fabs(x_46_im))
	else:
		tmp = math.fabs(x_46_im) * (x_46_re * ((x_46_re + x_46_re) - ((math.fabs(x_46_im) / x_46_re) * (math.fabs(x_46_im) - x_46_re))))
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 1.45e+93)
		tmp = Float64(Float64(3.0 * Float64(Float64(abs(x_46_im) * x_46_re) * x_46_re)) - Float64(Float64(abs(x_46_im) * abs(x_46_im)) * abs(x_46_im)));
	else
		tmp = Float64(abs(x_46_im) * Float64(x_46_re * Float64(Float64(x_46_re + x_46_re) - Float64(Float64(abs(x_46_im) / x_46_re) * Float64(abs(x_46_im) - x_46_re)))));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (abs(x_46_im) <= 1.45e+93)
		tmp = (3.0 * ((abs(x_46_im) * x_46_re) * x_46_re)) - ((abs(x_46_im) * abs(x_46_im)) * abs(x_46_im));
	else
		tmp = abs(x_46_im) * (x_46_re * ((x_46_re + x_46_re) - ((abs(x_46_im) / x_46_re) * (abs(x_46_im) - x_46_re))));
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 1449999999999999918156969066552575526807194060320427185510591162486664329368057600973507395584], N[(N[(3 * N[(N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[(x$46$re * N[(N[(x$46$re + x$46$re), $MachinePrecision] - N[(N[(N[Abs[x$46$im], $MachinePrecision] / x$46$re), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.4499999999999999e93

   1. Initial program 82.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-neg-out N/A

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

      13. associate-+r+ N/A

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

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

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

      15. lower--.f64 N/A

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

   3. Applied rewrites 86.1%

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

   if 1.4499999999999999e93 < x.im

   1. Initial program 82.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. sub-flip-reverse N/A

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

   3. Applied rewrites 91.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-+.f64 N/A

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

      6. lower-unsound-/.f64 89.8%

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

   5. Applied rewrites 89.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.5%

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. add-to-fraction N/A

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

      14. add-flip N/A

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

      15. *-lft-identity N/A

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

      16. add-flip N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lower-/.f64 91.5%

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in x.re around 0

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

   10. Applied rewrites 76.2%

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

Alternative 3: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot \left|x.im\right|\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq \frac{7136238463529799}{713623846352979940529142984724747568191373312}:\\ \;\;\;\;3 \cdot \left(\left(\left|x.im\right| \cdot x.re\right) \cdot x.re\right) - t\_0 \cdot \left|x.im\right|\\ \mathbf{elif}\;\left|x.im\right| \leq 4199999999999999946706138771300051547218254797552787317160192849437054678070623108005076964146437683317590632178547399973133702187608219052664988969395368962445377968648764222603264:\\ \;\;\;\;\left|x.im\right| \cdot \left(\left(x.re + x.re\right) \cdot x.re - \left(\left|x.im\right| - x.re\right) \cdot \left(\left|x.im\right| + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-\left|x.im\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) (fabs x.im))))
  (*
   (copysign 1 x.im)
   (if (<=
        (fabs x.im)
        7136238463529799/713623846352979940529142984724747568191373312)
     (- (* 3 (* (* (fabs x.im) x.re) x.re)) (* t_0 (fabs x.im)))
     (if (<=
          (fabs x.im)
          4199999999999999946706138771300051547218254797552787317160192849437054678070623108005076964146437683317590632178547399973133702187608219052664988969395368962445377968648764222603264)
       (*
        (fabs x.im)
        (-
         (* (+ x.re x.re) x.re)
         (* (- (fabs x.im) x.re) (+ (fabs x.im) x.re))))
       (* t_0 (- (fabs x.im))))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * fabs(x_46_im);
	double tmp;
	if (fabs(x_46_im) <= 1e-29) {
		tmp = (3.0 * ((fabs(x_46_im) * x_46_re) * x_46_re)) - (t_0 * fabs(x_46_im));
	} else if (fabs(x_46_im) <= 4.2e+180) {
		tmp = fabs(x_46_im) * (((x_46_re + x_46_re) * x_46_re) - ((fabs(x_46_im) - x_46_re) * (fabs(x_46_im) + x_46_re)));
	} else {
		tmp = t_0 * -fabs(x_46_im);
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = Math.abs(x_46_im) * Math.abs(x_46_im);
	double tmp;
	if (Math.abs(x_46_im) <= 1e-29) {
		tmp = (3.0 * ((Math.abs(x_46_im) * x_46_re) * x_46_re)) - (t_0 * Math.abs(x_46_im));
	} else if (Math.abs(x_46_im) <= 4.2e+180) {
		tmp = Math.abs(x_46_im) * (((x_46_re + x_46_re) * x_46_re) - ((Math.abs(x_46_im) - x_46_re) * (Math.abs(x_46_im) + x_46_re)));
	} else {
		tmp = t_0 * -Math.abs(x_46_im);
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = math.fabs(x_46_im) * math.fabs(x_46_im)
	tmp = 0
	if math.fabs(x_46_im) <= 1e-29:
		tmp = (3.0 * ((math.fabs(x_46_im) * x_46_re) * x_46_re)) - (t_0 * math.fabs(x_46_im))
	elif math.fabs(x_46_im) <= 4.2e+180:
		tmp = math.fabs(x_46_im) * (((x_46_re + x_46_re) * x_46_re) - ((math.fabs(x_46_im) - x_46_re) * (math.fabs(x_46_im) + x_46_re)))
	else:
		tmp = t_0 * -math.fabs(x_46_im)
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * abs(x_46_im))
	tmp = 0.0
	if (abs(x_46_im) <= 1e-29)
		tmp = Float64(Float64(3.0 * Float64(Float64(abs(x_46_im) * x_46_re) * x_46_re)) - Float64(t_0 * abs(x_46_im)));
	elseif (abs(x_46_im) <= 4.2e+180)
		tmp = Float64(abs(x_46_im) * Float64(Float64(Float64(x_46_re + x_46_re) * x_46_re) - Float64(Float64(abs(x_46_im) - x_46_re) * Float64(abs(x_46_im) + x_46_re))));
	else
		tmp = Float64(t_0 * Float64(-abs(x_46_im)));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = abs(x_46_im) * abs(x_46_im);
	tmp = 0.0;
	if (abs(x_46_im) <= 1e-29)
		tmp = (3.0 * ((abs(x_46_im) * x_46_re) * x_46_re)) - (t_0 * abs(x_46_im));
	elseif (abs(x_46_im) <= 4.2e+180)
		tmp = abs(x_46_im) * (((x_46_re + x_46_re) * x_46_re) - ((abs(x_46_im) - x_46_re) * (abs(x_46_im) + x_46_re)));
	else
		tmp = t_0 * -abs(x_46_im);
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 7136238463529799/713623846352979940529142984724747568191373312], N[(N[(3 * N[(N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 4199999999999999946706138771300051547218254797552787317160192849437054678070623108005076964146437683317590632178547399973133702187608219052664988969395368962445377968648764222603264], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[Abs[x$46$im], $MachinePrecision] - x$46$re), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * (-N[Abs[x$46$im], $MachinePrecision])), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.im < 9.9999999999999994e-30

   1. Initial program 82.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-neg-out N/A

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

      13. associate-+r+ N/A

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

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

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

      15. lower--.f64 N/A

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

   3. Applied rewrites 86.1%

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

   if 9.9999999999999994e-30 < x.im < 4.1999999999999999e180

   1. Initial program 82.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. sub-flip-reverse N/A

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

   3. Applied rewrites 91.0%

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

   if 4.1999999999999999e180 < x.im

   1. Initial program 82.7%

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

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

      3. add-flip N/A

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

      4. sub-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 48.7%

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

   4. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. lower-pow.f64 58.6%

         \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

   6. Applied rewrites 58.6%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      4. unpow3 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 58.5%

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

   8. Applied rewrites 58.5%

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

Alternative 4: 96.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot \left|x.im\right|\\ t_1 := t\_0 \cdot \left(-\left|x.im\right|\right)\\ t_2 := \left(x.re \cdot x.re - t\_0\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + \left|x.im\right| \cdot x.re\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq \frac{-1265}{12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left(\left(\frac{3}{2} \cdot \left|x.im\right|\right) \cdot \left(x.re + x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) (fabs x.im)))
       (t_1 (* t_0 (- (fabs x.im))))
       (t_2
        (+
         (* (- (* x.re x.re) t_0) (fabs x.im))
         (* (+ (* x.re (fabs x.im)) (* (fabs x.im) x.re)) x.re))))
  (*
   (copysign 1 x.im)
   (if (<=
        t_2
        -1265/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424)
     t_1
     (if (<= t_2 INFINITY)
       (* (* (* 3/2 (fabs x.im)) (+ x.re x.re)) x.re)
       t_1)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * fabs(x_46_im);
	double t_1 = t_0 * -fabs(x_46_im);
	double t_2 = (((x_46_re * x_46_re) - t_0) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + (fabs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_2 <= -1e-319) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((1.5 * fabs(x_46_im)) * (x_46_re + x_46_re)) * x_46_re;
	} else {
		tmp = t_1;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = Math.abs(x_46_im) * Math.abs(x_46_im);
	double t_1 = t_0 * -Math.abs(x_46_im);
	double t_2 = (((x_46_re * x_46_re) - t_0) * Math.abs(x_46_im)) + (((x_46_re * Math.abs(x_46_im)) + (Math.abs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_2 <= -1e-319) {
		tmp = t_1;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = ((1.5 * Math.abs(x_46_im)) * (x_46_re + x_46_re)) * x_46_re;
	} else {
		tmp = t_1;
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = math.fabs(x_46_im) * math.fabs(x_46_im)
	t_1 = t_0 * -math.fabs(x_46_im)
	t_2 = (((x_46_re * x_46_re) - t_0) * math.fabs(x_46_im)) + (((x_46_re * math.fabs(x_46_im)) + (math.fabs(x_46_im) * x_46_re)) * x_46_re)
	tmp = 0
	if t_2 <= -1e-319:
		tmp = t_1
	elif t_2 <= math.inf:
		tmp = ((1.5 * math.fabs(x_46_im)) * (x_46_re + x_46_re)) * x_46_re
	else:
		tmp = t_1
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * abs(x_46_im))
	t_1 = Float64(t_0 * Float64(-abs(x_46_im)))
	t_2 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - t_0) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + Float64(abs(x_46_im) * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_2 <= -1e-319)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(1.5 * abs(x_46_im)) * Float64(x_46_re + x_46_re)) * x_46_re);
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = abs(x_46_im) * abs(x_46_im);
	t_1 = t_0 * -abs(x_46_im);
	t_2 = (((x_46_re * x_46_re) - t_0) * abs(x_46_im)) + (((x_46_re * abs(x_46_im)) + (abs(x_46_im) * x_46_re)) * x_46_re);
	tmp = 0.0;
	if (t_2 <= -1e-319)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = ((1.5 * abs(x_46_im)) * (x_46_re + x_46_re)) * x_46_re;
	else
		tmp = t_1;
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * (-N[Abs[x$46$im], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - t$95$0), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1265/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424], t$95$1, If[LessEqual[t$95$2, Infinity], N[(N[(N[(3/2 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], t$95$1]]), $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)) < -9.9998886718268301e-320 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 82.7%

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

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

      3. add-flip N/A

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

      4. sub-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 48.7%

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

   4. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. lower-pow.f64 58.6%

         \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

   6. Applied rewrites 58.6%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      4. unpow3 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 58.5%

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

   8. Applied rewrites 58.5%

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

   if -9.9998886718268301e-320 < (+.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 82.7%

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

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

      3. add-flip N/A

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

      4. sub-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 48.7%

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

   4. Taylor expanded in x.re around inf

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

   6. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 56.3%

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

   8. Applied rewrites 56.3%

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

Alternative 5: 96.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot \left|x.im\right|\\ t_1 := t\_0 \cdot \left(-\left|x.im\right|\right)\\ t_2 := \left(x.re \cdot x.re - t\_0\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + \left|x.im\right| \cdot x.re\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq \frac{-1265}{12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{3}{2} \cdot \left(\left(\left(x.re + x.re\right) \cdot \left|x.im\right|\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) (fabs x.im)))
       (t_1 (* t_0 (- (fabs x.im))))
       (t_2
        (+
         (* (- (* x.re x.re) t_0) (fabs x.im))
         (* (+ (* x.re (fabs x.im)) (* (fabs x.im) x.re)) x.re))))
  (*
   (copysign 1 x.im)
   (if (<=
        t_2
        -1265/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424)
     t_1
     (if (<= t_2 INFINITY)
       (* 3/2 (* (* (+ x.re x.re) (fabs x.im)) x.re))
       t_1)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * fabs(x_46_im);
	double t_1 = t_0 * -fabs(x_46_im);
	double t_2 = (((x_46_re * x_46_re) - t_0) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + (fabs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_2 <= -1e-319) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 1.5 * (((x_46_re + x_46_re) * fabs(x_46_im)) * x_46_re);
	} else {
		tmp = t_1;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = Math.abs(x_46_im) * Math.abs(x_46_im);
	double t_1 = t_0 * -Math.abs(x_46_im);
	double t_2 = (((x_46_re * x_46_re) - t_0) * Math.abs(x_46_im)) + (((x_46_re * Math.abs(x_46_im)) + (Math.abs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_2 <= -1e-319) {
		tmp = t_1;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = 1.5 * (((x_46_re + x_46_re) * Math.abs(x_46_im)) * x_46_re);
	} else {
		tmp = t_1;
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = math.fabs(x_46_im) * math.fabs(x_46_im)
	t_1 = t_0 * -math.fabs(x_46_im)
	t_2 = (((x_46_re * x_46_re) - t_0) * math.fabs(x_46_im)) + (((x_46_re * math.fabs(x_46_im)) + (math.fabs(x_46_im) * x_46_re)) * x_46_re)
	tmp = 0
	if t_2 <= -1e-319:
		tmp = t_1
	elif t_2 <= math.inf:
		tmp = 1.5 * (((x_46_re + x_46_re) * math.fabs(x_46_im)) * x_46_re)
	else:
		tmp = t_1
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * abs(x_46_im))
	t_1 = Float64(t_0 * Float64(-abs(x_46_im)))
	t_2 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - t_0) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + Float64(abs(x_46_im) * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_2 <= -1e-319)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(1.5 * Float64(Float64(Float64(x_46_re + x_46_re) * abs(x_46_im)) * x_46_re));
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = abs(x_46_im) * abs(x_46_im);
	t_1 = t_0 * -abs(x_46_im);
	t_2 = (((x_46_re * x_46_re) - t_0) * abs(x_46_im)) + (((x_46_re * abs(x_46_im)) + (abs(x_46_im) * x_46_re)) * x_46_re);
	tmp = 0.0;
	if (t_2 <= -1e-319)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = 1.5 * (((x_46_re + x_46_re) * abs(x_46_im)) * x_46_re);
	else
		tmp = t_1;
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * (-N[Abs[x$46$im], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - t$95$0), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1265/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424], t$95$1, If[LessEqual[t$95$2, Infinity], N[(3/2 * N[(N[(N[(x$46$re + x$46$re), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], t$95$1]]), $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)) < -9.9998886718268301e-320 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 82.7%

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

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

      3. add-flip N/A

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

      4. sub-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 48.7%

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

   4. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. lower-pow.f64 58.6%

         \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

   6. Applied rewrites 58.6%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      4. unpow3 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 58.5%

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

   8. Applied rewrites 58.5%

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

   if -9.9998886718268301e-320 < (+.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 82.7%

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

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

      3. add-flip N/A

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

      4. sub-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 48.7%

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

   4. Taylor expanded in x.re around inf

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

   6. Applied rewrites 56.3%

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

Alternative 6: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| + \left|x.re\right|\\ \mathbf{if}\;\left|x.re\right| \leq 7200000000000000146937072365426998147461396793116072776883800022538200438210643876239948474488778237998346713711784938990991102441042258372844127730532352:\\ \;\;\;\;x.im \cdot \left(t\_0 \cdot \left|x.re\right| - \left(x.im - \left|x.re\right|\right) \cdot \left(x.im + \left|x.re\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{2} \cdot \left(\left(t\_0 \cdot x.im\right) \cdot \left|x.re\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (+ (fabs x.re) (fabs x.re))))
  (if (<=
       (fabs x.re)
       7200000000000000146937072365426998147461396793116072776883800022538200438210643876239948474488778237998346713711784938990991102441042258372844127730532352)
    (*
     x.im
     (-
      (* t_0 (fabs x.re))
      (* (- x.im (fabs x.re)) (+ x.im (fabs x.re)))))
    (* 3/2 (* (* t_0 x.im) (fabs x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) + fabs(x_46_re);
	double tmp;
	if (fabs(x_46_re) <= 7.2e+153) {
		tmp = x_46_im * ((t_0 * fabs(x_46_re)) - ((x_46_im - fabs(x_46_re)) * (x_46_im + fabs(x_46_re))));
	} else {
		tmp = 1.5 * ((t_0 * x_46_im) * fabs(x_46_re));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(x_46re) + abs(x_46re)
    if (abs(x_46re) <= 7.2d+153) then
        tmp = x_46im * ((t_0 * abs(x_46re)) - ((x_46im - abs(x_46re)) * (x_46im + abs(x_46re))))
    else
        tmp = 1.5d0 * ((t_0 * x_46im) * abs(x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = Math.abs(x_46_re) + Math.abs(x_46_re);
	double tmp;
	if (Math.abs(x_46_re) <= 7.2e+153) {
		tmp = x_46_im * ((t_0 * Math.abs(x_46_re)) - ((x_46_im - Math.abs(x_46_re)) * (x_46_im + Math.abs(x_46_re))));
	} else {
		tmp = 1.5 * ((t_0 * x_46_im) * Math.abs(x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = math.fabs(x_46_re) + math.fabs(x_46_re)
	tmp = 0
	if math.fabs(x_46_re) <= 7.2e+153:
		tmp = x_46_im * ((t_0 * math.fabs(x_46_re)) - ((x_46_im - math.fabs(x_46_re)) * (x_46_im + math.fabs(x_46_re))))
	else:
		tmp = 1.5 * ((t_0 * x_46_im) * math.fabs(x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_re) + abs(x_46_re))
	tmp = 0.0
	if (abs(x_46_re) <= 7.2e+153)
		tmp = Float64(x_46_im * Float64(Float64(t_0 * abs(x_46_re)) - Float64(Float64(x_46_im - abs(x_46_re)) * Float64(x_46_im + abs(x_46_re)))));
	else
		tmp = Float64(1.5 * Float64(Float64(t_0 * x_46_im) * abs(x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = abs(x_46_re) + abs(x_46_re);
	tmp = 0.0;
	if (abs(x_46_re) <= 7.2e+153)
		tmp = x_46_im * ((t_0 * abs(x_46_re)) - ((x_46_im - abs(x_46_re)) * (x_46_im + abs(x_46_re))));
	else
		tmp = 1.5 * ((t_0 * x_46_im) * abs(x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 7200000000000000146937072365426998147461396793116072776883800022538200438210643876239948474488778237998346713711784938990991102441042258372844127730532352], N[(x$46$im * N[(N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$im - N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[(x$46$im + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3/2 * N[(N[(t$95$0 * x$46$im), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.2000000000000001e153

   1. Initial program 82.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. sub-flip-reverse N/A

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

   3. Applied rewrites 91.0%

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

   if 7.2000000000000001e153 < x.re

   1. Initial program 82.7%

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

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

      3. add-flip N/A

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

      4. sub-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 48.7%

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

   4. Taylor expanded in x.re around inf

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

   6. Applied rewrites 56.3%

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

Alternative 7: 90.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot \left|x.im\right|\\ t_1 := t\_0 \cdot \left(-\left|x.im\right|\right)\\ t_2 := \left(x.re \cdot x.re - t\_0\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + \left|x.im\right| \cdot x.re\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq \frac{-1265}{12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left|x.im\right| \cdot \left(x.re \cdot \left(3 \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) (fabs x.im)))
       (t_1 (* t_0 (- (fabs x.im))))
       (t_2
        (+
         (* (- (* x.re x.re) t_0) (fabs x.im))
         (* (+ (* x.re (fabs x.im)) (* (fabs x.im) x.re)) x.re))))
  (*
   (copysign 1 x.im)
   (if (<=
        t_2
        -1265/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424)
     t_1
     (if (<= t_2 INFINITY) (* (fabs x.im) (* x.re (* 3 x.re))) t_1)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * fabs(x_46_im);
	double t_1 = t_0 * -fabs(x_46_im);
	double t_2 = (((x_46_re * x_46_re) - t_0) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + (fabs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_2 <= -1e-319) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fabs(x_46_im) * (x_46_re * (3.0 * x_46_re));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = Math.abs(x_46_im) * Math.abs(x_46_im);
	double t_1 = t_0 * -Math.abs(x_46_im);
	double t_2 = (((x_46_re * x_46_re) - t_0) * Math.abs(x_46_im)) + (((x_46_re * Math.abs(x_46_im)) + (Math.abs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_2 <= -1e-319) {
		tmp = t_1;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.abs(x_46_im) * (x_46_re * (3.0 * x_46_re));
	} else {
		tmp = t_1;
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = math.fabs(x_46_im) * math.fabs(x_46_im)
	t_1 = t_0 * -math.fabs(x_46_im)
	t_2 = (((x_46_re * x_46_re) - t_0) * math.fabs(x_46_im)) + (((x_46_re * math.fabs(x_46_im)) + (math.fabs(x_46_im) * x_46_re)) * x_46_re)
	tmp = 0
	if t_2 <= -1e-319:
		tmp = t_1
	elif t_2 <= math.inf:
		tmp = math.fabs(x_46_im) * (x_46_re * (3.0 * x_46_re))
	else:
		tmp = t_1
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * abs(x_46_im))
	t_1 = Float64(t_0 * Float64(-abs(x_46_im)))
	t_2 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - t_0) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + Float64(abs(x_46_im) * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_2 <= -1e-319)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(abs(x_46_im) * Float64(x_46_re * Float64(3.0 * x_46_re)));
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = abs(x_46_im) * abs(x_46_im);
	t_1 = t_0 * -abs(x_46_im);
	t_2 = (((x_46_re * x_46_re) - t_0) * abs(x_46_im)) + (((x_46_re * abs(x_46_im)) + (abs(x_46_im) * x_46_re)) * x_46_re);
	tmp = 0.0;
	if (t_2 <= -1e-319)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = abs(x_46_im) * (x_46_re * (3.0 * x_46_re));
	else
		tmp = t_1;
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * (-N[Abs[x$46$im], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - t$95$0), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1265/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424], t$95$1, If[LessEqual[t$95$2, Infinity], N[(N[Abs[x$46$im], $MachinePrecision] * N[(x$46$re * N[(3 * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $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)) < -9.9998886718268301e-320 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 82.7%

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

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

      3. add-flip N/A

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

      4. sub-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 48.7%

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

   4. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. lower-pow.f64 58.6%

         \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

   6. Applied rewrites 58.6%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      4. unpow3 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 58.5%

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

   8. Applied rewrites 58.5%

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

   if -9.9998886718268301e-320 < (+.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 82.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. sub-flip-reverse N/A

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

   3. Applied rewrites 91.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-+.f64 N/A

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

      6. lower-unsound-/.f64 89.8%

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

   5. Applied rewrites 89.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.5%

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. add-to-fraction N/A

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

      14. add-flip N/A

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

      15. *-lft-identity N/A

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

      16. add-flip N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lower-/.f64 91.5%

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in x.re around inf

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

      1. lower-*.f64 50.7%

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

   10. Applied rewrites 50.7%

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

Alternative 8: 58.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return (x_46_im * x_46_im) * -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_46im * x_46im) * -x_46im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

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

   3. add-flip N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

3. Applied rewrites 48.7%

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

4. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

   2. lower-pow.f64 58.6%

      \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

6. Applied rewrites 58.6%

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

   4. unpow3 N/A

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

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

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 58.5%

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

8. Applied rewrites 58.5%

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

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64
  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))