math.cube on complex, real part

Percentage Accurate: 82.2% → 99.3%

Time: 7.9s

Alternatives: 11

Speedup: 0.7×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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_46re) - (((x_46re * x_46im) + (x_46im * x_46re)) * x_46im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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_46re) - (((x_46re * x_46im) + (x_46im * x_46re)) * x_46im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(-3 \cdot x.im\_m\right) \cdot x.re\_m\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im\_m \cdot x.im\_m, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re\_m - x.im\_m\right) \cdot x.re\_m, x.im\_m + x.re\_m, x.im\_m\right) + x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))))
   (*
    x.re_s
    (if (<= t_0 -5e+90)
      (* (* (* -3.0 x.im_m) x.re_m) x.im_m)
      (if (<= t_0 5e-114)
        (* (fma -3.0 (* x.im_m x.im_m) (* x.re_m x.re_m)) x.re_m)
        (+
         (fma (* (- x.re_m x.im_m) x.re_m) (+ x.im_m x.re_m) x.im_m)
         x.im_m))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= -5e+90) {
		tmp = ((-3.0 * x_46_im_m) * x_46_re_m) * x_46_im_m;
	} else if (t_0 <= 5e-114) {
		tmp = fma(-3.0, (x_46_im_m * x_46_im_m), (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = fma(((x_46_re_m - x_46_im_m) * x_46_re_m), (x_46_im_m + x_46_re_m), x_46_im_m) + x_46_im_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= -5e+90)
		tmp = Float64(Float64(Float64(-3.0 * x_46_im_m) * x_46_re_m) * x_46_im_m);
	elseif (t_0 <= 5e-114)
		tmp = Float64(fma(-3.0, Float64(x_46_im_m * x_46_im_m), Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = Float64(fma(Float64(Float64(x_46_re_m - x_46_im_m) * x_46_re_m), Float64(x_46_im_m + x_46_re_m), x_46_im_m) + x_46_im_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -5e+90], N[(N[(N[(-3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e-114], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] + x$46$im$95$m), $MachinePrecision] + x$46$im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -5.0000000000000004e90

   1. Initial program 87.5%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   7. Applied rewrites 41.6%

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

   if -5.0000000000000004e90 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 4.99999999999999989e-114

   1. Initial program 99.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   if 4.99999999999999989e-114 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 83.9

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

   6. Applied rewrites 83.9%

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

   8. Applied rewrites 82.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 82.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 82.8

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

   10. Applied rewrites 82.8%

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

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(-3 \cdot x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re\_m - x.im\_m\right) \cdot x.re\_m, x.im\_m + x.re\_m, x.im\_m\right) + x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))))
   (*
    x.re_s
    (if (<= t_0 -5e-308)
      (* (* x.re_m x.im_m) (* -3.0 x.im_m))
      (if (<= t_0 5e-114)
        (* (* x.re_m x.re_m) x.re_m)
        (+
         (fma (* (- x.re_m x.im_m) x.re_m) (+ x.im_m x.re_m) x.im_m)
         x.im_m))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = (x_46_re_m * x_46_im_m) * (-3.0 * x_46_im_m);
	} else if (t_0 <= 5e-114) {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	} else {
		tmp = fma(((x_46_re_m - x_46_im_m) * x_46_re_m), (x_46_im_m + x_46_re_m), x_46_im_m) + x_46_im_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(-3.0 * x_46_im_m));
	elseif (t_0 <= 5e-114)
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	else
		tmp = Float64(fma(Float64(Float64(x_46_re_m - x_46_im_m) * x_46_re_m), Float64(x_46_im_m + x_46_re_m), x_46_im_m) + x_46_im_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -5e-308], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(-3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-114], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] + x$46$im$95$m), $MachinePrecision] + x$46$im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.99999999999999955e-308

   1. Initial program 91.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 41.4

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

   5. Applied rewrites 41.4%

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

   7. Applied rewrites 41.4%

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

   if -4.99999999999999955e-308 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 4.99999999999999989e-114

   1. Initial program 99.8%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 77.3%

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

   if 4.99999999999999989e-114 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 83.9

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

   6. Applied rewrites 83.9%

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

   8. Applied rewrites 82.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 82.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 82.8

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

   10. Applied rewrites 82.8%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\left(x.im\_m + x.re\_m\right) \cdot \left(\mathsf{fma}\left(\frac{x.re\_m}{x.im\_m}, x.re\_m, -x.re\_m\right) \cdot x.im\_m\right) - \left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re\_m - x.im\_m\right) \cdot x.re\_m, x.im\_m + x.re\_m, x.im\_m\right) + x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       5e-114)
    (-
     (* (+ x.im_m x.re_m) (* (fma (/ x.re_m x.im_m) x.re_m (- x.re_m)) x.im_m))
     (* (* x.re_m (+ x.im_m x.im_m)) x.im_m))
    (+ (fma (* (- x.re_m x.im_m) x.re_m) (+ x.im_m x.re_m) x.im_m) x.im_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-114) {
		tmp = ((x_46_im_m + x_46_re_m) * (fma((x_46_re_m / x_46_im_m), x_46_re_m, -x_46_re_m) * x_46_im_m)) - ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_im_m);
	} else {
		tmp = fma(((x_46_re_m - x_46_im_m) * x_46_re_m), (x_46_im_m + x_46_re_m), x_46_im_m) + x_46_im_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-114)
		tmp = Float64(Float64(Float64(x_46_im_m + x_46_re_m) * Float64(fma(Float64(x_46_re_m / x_46_im_m), x_46_re_m, Float64(-x_46_re_m)) * x_46_im_m)) - Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)) * x_46_im_m));
	else
		tmp = Float64(fma(Float64(Float64(x_46_re_m - x_46_im_m) * x_46_re_m), Float64(x_46_im_m + x_46_re_m), x_46_im_m) + x_46_im_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], 5e-114], N[(N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * N[(N[(N[(x$46$re$95$m / x$46$im$95$m), $MachinePrecision] * x$46$re$95$m + (-x$46$re$95$m)), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] + x$46$im$95$m), $MachinePrecision] + x$46$im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 4.99999999999999989e-114

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x.im around inf

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

      1. distribute-lft-in N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 97.8%

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

   if 4.99999999999999989e-114 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 83.9

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

   6. Applied rewrites 83.9%

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

   8. Applied rewrites 82.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 82.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 82.8

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

   10. Applied rewrites 82.8%

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

Alternative 4: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m - x.im\_m\right) \cdot x.re\_m\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\left(x.im\_m + x.re\_m\right) \cdot t\_0 - \left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x.im\_m + x.re\_m, x.im\_m\right) + x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* (- x.re_m x.im_m) x.re_m)))
   (*
    x.re_s
    (if (<=
         (-
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
         5e-114)
      (- (* (+ x.im_m x.re_m) t_0) (* (* x.re_m (+ x.im_m x.im_m)) x.im_m))
      (+ (fma t_0 (+ x.im_m x.re_m) x.im_m) x.im_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (x_46_re_m - x_46_im_m) * x_46_re_m;
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-114) {
		tmp = ((x_46_im_m + x_46_re_m) * t_0) - ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_im_m);
	} else {
		tmp = fma(t_0, (x_46_im_m + x_46_re_m), x_46_im_m) + x_46_im_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(x_46_re_m - x_46_im_m) * x_46_re_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-114)
		tmp = Float64(Float64(Float64(x_46_im_m + x_46_re_m) * t_0) - Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)) * x_46_im_m));
	else
		tmp = Float64(fma(t_0, Float64(x_46_im_m + x_46_re_m), x_46_im_m) + x_46_im_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], 5e-114], N[(N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] + x$46$im$95$m), $MachinePrecision] + x$46$im$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 4.99999999999999989e-114

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if 4.99999999999999989e-114 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 83.9

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

   6. Applied rewrites 83.9%

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

   8. Applied rewrites 82.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 82.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 82.8

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

   10. Applied rewrites 82.8%

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

Alternative 5: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(-3 \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       -5e-308)
    (* (* x.re_m x.im_m) (* -3.0 x.im_m))
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308) {
		tmp = (x_46_re_m * x_46_im_m) * (-3.0 * x_46_im_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

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

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

real(8) function code(x_46re_s, x_46re_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46re_m * x_46im_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-5d-308)) then
        tmp = (x_46re_m * x_46im_m) * ((-3.0d0) * x_46im_m)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308) {
		tmp = (x_46_re_m * x_46_im_m) * (-3.0 * x_46_im_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308:
		tmp = (x_46_re_m * x_46_im_m) * (-3.0 * x_46_im_m)
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308)
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(-3.0 * x_46_im_m));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308)
		tmp = (x_46_re_m * x_46_im_m) * (-3.0 * x_46_im_m);
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -5e-308], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(-3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.99999999999999955e-308

   1. Initial program 91.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 41.4

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

   5. Applied rewrites 41.4%

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

   7. Applied rewrites 41.4%

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

   if -4.99999999999999955e-308 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 78.9%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 63.8%

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

Alternative 6: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -5 \cdot 10^{-308}:\\ \;\;\;\;-3 \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       -5e-308)
    (* -3.0 (* (* x.im_m x.re_m) x.im_m))
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308) {
		tmp = -3.0 * ((x_46_im_m * x_46_re_m) * x_46_im_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

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

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

real(8) function code(x_46re_s, x_46re_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46re_m * x_46im_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-5d-308)) then
        tmp = (-3.0d0) * ((x_46im_m * x_46re_m) * x_46im_m)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308) {
		tmp = -3.0 * ((x_46_im_m * x_46_re_m) * x_46_im_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308:
		tmp = -3.0 * ((x_46_im_m * x_46_re_m) * x_46_im_m)
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im_m * x_46_re_m) * x_46_im_m));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308)
		tmp = -3.0 * ((x_46_im_m * x_46_re_m) * x_46_im_m);
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -5e-308], N[(-3.0 * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.99999999999999955e-308

   1. Initial program 91.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 41.4

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

   5. Applied rewrites 41.4%

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

   if -4.99999999999999955e-308 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 78.9%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 63.8%

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

Alternative 7: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -5 \cdot 10^{-308}:\\ \;\;\;\;-3 \cdot \left(\left(x.im\_m \cdot x.im\_m\right) \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       -5e-308)
    (* -3.0 (* (* x.im_m x.im_m) x.re_m))
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308) {
		tmp = -3.0 * ((x_46_im_m * x_46_im_m) * x_46_re_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

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

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

real(8) function code(x_46re_s, x_46re_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46re_m * x_46im_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-5d-308)) then
        tmp = (-3.0d0) * ((x_46im_m * x_46im_m) * x_46re_m)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308) {
		tmp = -3.0 * ((x_46_im_m * x_46_im_m) * x_46_re_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308:
		tmp = -3.0 * ((x_46_im_m * x_46_im_m) * x_46_re_m)
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im_m * x_46_im_m) * x_46_re_m));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -5e-308)
		tmp = -3.0 * ((x_46_im_m * x_46_im_m) * x_46_re_m);
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -5e-308], N[(-3.0 * N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.99999999999999955e-308

   1. Initial program 91.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 41.4

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

   5. Applied rewrites 41.4%

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

   7. Applied rewrites 33.4%

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

   if -4.99999999999999955e-308 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 78.9%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 63.8%

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

Alternative 8: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -4 \cdot 10^{-260}:\\ \;\;\;\;\left(x.re\_m \cdot \left(2 - x.im\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       -4e-260)
    (* (* x.re_m (- 2.0 x.im_m)) x.im_m)
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -4e-260) {
		tmp = (x_46_re_m * (2.0 - x_46_im_m)) * x_46_im_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

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

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

real(8) function code(x_46re_s, x_46re_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46re_m * x_46im_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-4d-260)) then
        tmp = (x_46re_m * (2.0d0 - x_46im_m)) * x_46im_m
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -4e-260) {
		tmp = (x_46_re_m * (2.0 - x_46_im_m)) * x_46_im_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -4e-260:
		tmp = (x_46_re_m * (2.0 - x_46_im_m)) * x_46_im_m
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -4e-260)
		tmp = Float64(Float64(x_46_re_m * Float64(2.0 - x_46_im_m)) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -4e-260)
		tmp = (x_46_re_m * (2.0 - x_46_im_m)) * x_46_im_m;
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-260], N[(N[(x$46$re$95$m * N[(2.0 - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -3.99999999999999985e-260

   1. Initial program 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x.im around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 97.9

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

   7. Applied rewrites 97.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   9. Applied rewrites 71.0%

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

   10. Taylor expanded in x.re around 0

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

   12. Applied rewrites 17.4%

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

   if -3.99999999999999985e-260 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 79.0%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 63.4%

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

Alternative 9: 62.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.8 \cdot 10^{+186}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.re\_m\right) \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 2.8e+186)
    (* (* x.re_m x.re_m) x.re_m)
    (* (* (- x.re_m) x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.8e+186) {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	} else {
		tmp = (-x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

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

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

real(8) function code(x_46re_s, x_46re_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 2.8d+186) then
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    else
        tmp = (-x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.8e+186) {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	} else {
		tmp = (-x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 2.8e+186:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	else:
		tmp = (-x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2.8e+186)
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	else
		tmp = Float64(Float64(Float64(-x_46_re_m) * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 2.8e+186)
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	else
		tmp = (-x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 2.8e+186], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[((-x$46$re$95$m) * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.80000000000000018e186

   1. Initial program 87.8%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 67.8%

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

   if 2.80000000000000018e186 < x.im

   1. Initial program 51.9%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 12.4

         \[\leadsto \color{blue}{{x.re}^{3}} \]

   5. Applied rewrites 12.4%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 17.6%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.8 \cdot 10^{+186}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.re\right) \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_46re_s, x_46re_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * ((x_46re_m * x_46re_m) * x_46re_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in x.re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 88.1%

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

6. Taylor expanded in x.re around inf

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

8. Applied rewrites 62.1%

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

Alternative 11: 2.8% accurate, 6.7× speedup

Math

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

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m) :precision binary64 (* x.re_s (* 2.0 x.im_m)))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (2.0 * x_46_im_m);
}

Fortran

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

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

real(8) function code(x_46re_s, x_46re_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * (2.0d0 * x_46im_m)
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (2.0 * x_46_im_m);
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * (2.0 * x_46_im_m)

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(2.0 * x_46_im_m))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * (2.0 * x_46_im_m);
end

Wolfram

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

Derivation

1. Initial program 84.2%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. difference-of-squares N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 93.1

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

4. Applied rewrites 93.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 93.1

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

6. Applied rewrites 93.1%

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

8. Applied rewrites 63.2%

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

9. Taylor expanded in x.re around 0

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

    1. lower-*.f64 3.4

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

11. Applied rewrites 3.4%

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

Developer Target 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Fortran

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_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024363 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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