math.cube on complex, real part

Percentage Accurate: 82.5% → 97.8%

Time: 4.8s

Alternatives: 8

Speedup: 1.4×

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

Specification

Math

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

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: 82.5% 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: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 8 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re\_m, \left(x.re\_m - x.im\right) \cdot x.re\_m, \left(-2 \cdot \left(x.im \cdot x.re\_m\right)\right) \cdot x.im\right)\\ \mathbf{elif}\;x.re\_m \leq 1.8 \cdot 10^{+231}:\\ \;\;\;\;\left(\mathsf{fma}\left(x.re\_m, x.re\_m, \left(-x.im\right) \cdot x.im\right) - \left(x.im \cdot x.im\right) \cdot 2\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 8e-91)
    (fma
     (+ x.im x.re_m)
     (* (- x.re_m x.im) x.re_m)
     (* (* -2.0 (* x.im x.re_m)) x.im))
    (if (<= x.re_m 1.8e+231)
      (*
       (- (fma x.re_m x.re_m (* (- x.im) x.im)) (* (* x.im x.im) 2.0))
       x.re_m)
      (* (* x.re_m x.re_m) x.re_m)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 8e-91) {
		tmp = fma((x_46_im + x_46_re_m), ((x_46_re_m - x_46_im) * x_46_re_m), ((-2.0 * (x_46_im * x_46_re_m)) * x_46_im));
	} else if (x_46_re_m <= 1.8e+231) {
		tmp = (fma(x_46_re_m, x_46_re_m, (-x_46_im * x_46_im)) - ((x_46_im * x_46_im) * 2.0)) * x_46_re_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 8e-91)
		tmp = fma(Float64(x_46_im + x_46_re_m), Float64(Float64(x_46_re_m - x_46_im) * x_46_re_m), Float64(Float64(-2.0 * Float64(x_46_im * x_46_re_m)) * x_46_im));
	elseif (x_46_re_m <= 1.8e+231)
		tmp = Float64(Float64(fma(x_46_re_m, x_46_re_m, Float64(Float64(-x_46_im) * x_46_im)) - Float64(Float64(x_46_im * x_46_im) * 2.0)) * 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

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 8e-91], N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + N[(N[(-2.0 * N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.8e+231], N[(N[(N[(x$46$re$95$m * x$46$re$95$m + N[((-x$46$im) * x$46$im), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$im * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $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 3 regimes

2. if x.re < 8.00000000000000018e-91

   1. Initial program 84.8%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 86.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 95.2

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

   6. Applied rewrites 95.2%

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

   if 8.00000000000000018e-91 < x.re < 1.8e231

   1. Initial program 88.5%

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

   3. 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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.8e231 < x.re

   1. Initial program 61.5%

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

   3. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

         \[\leadsto {x.re}^{2} \cdot x.re \]

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 8 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re, \left(x.re - x.im\right) \cdot x.re, \left(-2 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.8 \cdot 10^{+231}:\\ \;\;\;\;\left(\mathsf{fma}\left(x.re, x.re, \left(-x.im\right) \cdot x.im\right) - \left(x.im \cdot x.im\right) \cdot 2\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(-3 \cdot x.re\_m\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       -2e-309)
    (* (* (* -3.0 x.re_m) x.im) x.im)
    (* (* x.re_m x.re_m) x.re_m))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309) {
		tmp = ((-3.0 * x_46_re_m) * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m =     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)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46re_m * x_46im) + (x_46im * x_46re_m)) * x_46im)) <= (-2d-309)) then
        tmp = (((-3.0d0) * x_46re_m) * x_46im) * x_46im
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309) {
		tmp = ((-3.0 * x_46_re_m) * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309:
		tmp = ((-3.0 * x_46_re_m) * x_46_im) * x_46_im
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309)
		tmp = Float64(Float64(Float64(-3.0 * x_46_re_m) * x_46_im) * x_46_im);
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309)
		tmp = ((-3.0 * x_46_re_m) * x_46_im) * x_46_im;
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e-309], N[(N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1.9999999999999988e-309

   1. Initial program 93.5%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 49.5

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

   5. Applied rewrites 49.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

         \[\leadsto \left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im \]

      6. metadata-eval N/A

         \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(-2 + -1\right)\right)\right) \cdot x.im \]

      7. distribute-rgt-out N/A

         \[\leadsto \left(x.im \cdot \left(-2 \cdot x.re + -1 \cdot x.re\right)\right) \cdot x.im \]

      8. lower-*.f64 N/A

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

      9. distribute-rgt-out N/A

         \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(-2 + -1\right)\right)\right) \cdot x.im \]

      10. metadata-eval N/A

          \[\leadsto \left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im \]

      12. *-commutative N/A

          \[\leadsto \left(\left(x.re \cdot -3\right) \cdot x.im\right) \cdot x.im \]

      13. lower-*.f64 55.9

          \[\leadsto \left(\left(x.re \cdot -3\right) \cdot x.im\right) \cdot x.im \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(x.re \cdot -3\right) \cdot x.im\right) \cdot x.im \]

      15. *-commutative N/A

          \[\leadsto \left(\left(-3 \cdot x.re\right) \cdot x.im\right) \cdot x.im \]

      16. lower-*.f64 55.9

          \[\leadsto \left(\left(-3 \cdot x.re\right) \cdot x.im\right) \cdot x.im \]

   7. Applied rewrites 55.9%

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

   if -1.9999999999999988e-309 < (-.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 78.8%

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

   3. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 57.4

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

   5. Applied rewrites 57.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

         \[\leadsto {x.re}^{2} \cdot x.re \]

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

Alternative 3: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       -2e-309)
    (* (* x.im (* x.re_m x.im)) -3.0)
    (* (* x.re_m x.re_m) x.re_m))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309) {
		tmp = (x_46_im * (x_46_re_m * x_46_im)) * -3.0;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m =     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)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46re_m * x_46im) + (x_46im * x_46re_m)) * x_46im)) <= (-2d-309)) then
        tmp = (x_46im * (x_46re_m * x_46im)) * (-3.0d0)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309) {
		tmp = (x_46_im * (x_46_re_m * x_46_im)) * -3.0;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309:
		tmp = (x_46_im * (x_46_re_m * x_46_im)) * -3.0
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309)
		tmp = Float64(Float64(x_46_im * Float64(x_46_re_m * x_46_im)) * -3.0);
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309)
		tmp = (x_46_im * (x_46_re_m * x_46_im)) * -3.0;
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e-309], N[(N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1.9999999999999988e-309

   1. Initial program 93.5%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 49.5

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

   5. Applied rewrites 49.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3 \]

      12. lift-*.f64 49.6

          \[\leadsto \left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3 \]

   7. Applied rewrites 49.6%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3 \]

      3. associate-*l* N/A

         \[\leadsto \left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3 \]

      5. *-commutative N/A

         \[\leadsto \left(x.im \cdot \left(x.re \cdot x.im\right)\right) \cdot -3 \]

      6. lower-*.f64 55.8

         \[\leadsto \left(x.im \cdot \left(x.re \cdot x.im\right)\right) \cdot -3 \]

   9. Applied rewrites 55.8%

      \[\leadsto \left(x.im \cdot \left(x.re \cdot x.im\right)\right) \cdot -3 \]

   if -1.9999999999999988e-309 < (-.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 78.8%

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

   3. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 57.4

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

   5. Applied rewrites 57.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

         \[\leadsto {x.re}^{2} \cdot x.re \]

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

Alternative 4: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot -3\right)\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       -2e-309)
    (* (* x.im (* x.im -3.0)) x.re_m)
    (* (* x.re_m x.re_m) x.re_m))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309) {
		tmp = (x_46_im * (x_46_im * -3.0)) * x_46_re_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m =     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)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46re_m * x_46im) + (x_46im * x_46re_m)) * x_46im)) <= (-2d-309)) then
        tmp = (x_46im * (x_46im * (-3.0d0))) * x_46re_m
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309) {
		tmp = (x_46_im * (x_46_im * -3.0)) * x_46_re_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309:
		tmp = (x_46_im * (x_46_im * -3.0)) * x_46_re_m
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309)
		tmp = Float64(Float64(x_46_im * Float64(x_46_im * -3.0)) * x_46_re_m);
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-309)
		tmp = (x_46_im * (x_46_im * -3.0)) * x_46_re_m;
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e-309], N[(N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $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 (-.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)) < -1.9999999999999988e-309

   1. Initial program 93.5%

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

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \left(\left(x.im \cdot x.im\right) \cdot \left(-1 - 2\right)\right) \cdot x.re \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(x.im \cdot x.im\right) \cdot \left(-1 - 2\right)\right) \cdot x.re \]

      7. metadata-eval 49.5

         \[\leadsto \left(\left(x.im \cdot x.im\right) \cdot -3\right) \cdot x.re \]

   5. Applied rewrites 49.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(x.im \cdot x.im\right) \cdot -3\right) \cdot x.re \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(x.im \cdot x.im\right) \cdot -3\right) \cdot x.re \]

      3. associate-*l* N/A

         \[\leadsto \left(x.im \cdot \left(x.im \cdot -3\right)\right) \cdot x.re \]

      4. lower-*.f64 N/A

         \[\leadsto \left(x.im \cdot \left(x.im \cdot -3\right)\right) \cdot x.re \]

      5. lower-*.f64 49.5

         \[\leadsto \left(x.im \cdot \left(x.im \cdot -3\right)\right) \cdot x.re \]

   7. Applied rewrites 49.5%

      \[\leadsto \left(x.im \cdot \left(x.im \cdot -3\right)\right) \cdot x.re \]

   if -1.9999999999999988e-309 < (-.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 78.8%

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

   3. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 57.4

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

   5. Applied rewrites 57.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

         \[\leadsto {x.re}^{2} \cdot x.re \]

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

Alternative 5: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.9 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \left(x.re\_m - x.im\right) \cdot x.re\_m, \left(-2 \cdot \left(x.im \cdot x.re\_m\right)\right) \cdot x.im\right)\\ \mathbf{elif}\;x.re\_m \leq 1.8 \cdot 10^{+231}:\\ \;\;\;\;\left(\mathsf{fma}\left(x.re\_m, x.re\_m, \left(-x.im\right) \cdot x.im\right) - \left(x.im \cdot x.im\right) \cdot 2\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1.9e-112)
    (fma x.im (* (- x.re_m x.im) x.re_m) (* (* -2.0 (* x.im x.re_m)) x.im))
    (if (<= x.re_m 1.8e+231)
      (*
       (- (fma x.re_m x.re_m (* (- x.im) x.im)) (* (* x.im x.im) 2.0))
       x.re_m)
      (* (* x.re_m x.re_m) x.re_m)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.9e-112) {
		tmp = fma(x_46_im, ((x_46_re_m - x_46_im) * x_46_re_m), ((-2.0 * (x_46_im * x_46_re_m)) * x_46_im));
	} else if (x_46_re_m <= 1.8e+231) {
		tmp = (fma(x_46_re_m, x_46_re_m, (-x_46_im * x_46_im)) - ((x_46_im * x_46_im) * 2.0)) * x_46_re_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1.9e-112)
		tmp = fma(x_46_im, Float64(Float64(x_46_re_m - x_46_im) * x_46_re_m), Float64(Float64(-2.0 * Float64(x_46_im * x_46_re_m)) * x_46_im));
	elseif (x_46_re_m <= 1.8e+231)
		tmp = Float64(Float64(fma(x_46_re_m, x_46_re_m, Float64(Float64(-x_46_im) * x_46_im)) - Float64(Float64(x_46_im * x_46_im) * 2.0)) * 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

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.9e-112], N[(x$46$im * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + N[(N[(-2.0 * N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.8e+231], N[(N[(N[(x$46$re$95$m * x$46$re$95$m + N[((-x$46$im) * x$46$im), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$im * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $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 3 regimes

2. if x.re < 1.89999999999999997e-112

   1. Initial program 84.6%

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

      1. lift--.f64 N/A

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

      2. 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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 86.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in x.re around 0

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

   9. Applied rewrites 74.2%

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

   if 1.89999999999999997e-112 < x.re < 1.8e231

   1. Initial program 88.5%

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

   3. 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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 1.8e231 < x.re

   1. Initial program 61.5%

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

   3. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

         \[\leadsto {x.re}^{2} \cdot x.re \]

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.9 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \left(x.re - x.im\right) \cdot x.re, \left(-2 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.8 \cdot 10^{+231}:\\ \;\;\;\;\left(\mathsf{fma}\left(x.re, x.re, \left(-x.im\right) \cdot x.im\right) - \left(x.im \cdot x.im\right) \cdot 2\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 5.4 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot x.im, -3, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \left(x.re\_m - x.im\right) \cdot x.re\_m, \left(-2 \cdot \left(x.im \cdot x.re\_m\right)\right) \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 5.4e+143)
    (* (fma (* x.im x.im) -3.0 (* x.re_m x.re_m)) x.re_m)
    (fma x.im (* (- x.re_m x.im) x.re_m) (* (* -2.0 (* x.im x.re_m)) x.im)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 5.4e+143) {
		tmp = fma((x_46_im * x_46_im), -3.0, (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = fma(x_46_im, ((x_46_re_m - x_46_im) * x_46_re_m), ((-2.0 * (x_46_im * x_46_re_m)) * x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 5.4e+143)
		tmp = Float64(fma(Float64(x_46_im * x_46_im), -3.0, Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = fma(x_46_im, Float64(Float64(x_46_re_m - x_46_im) * x_46_re_m), Float64(Float64(-2.0 * Float64(x_46_im * x_46_re_m)) * x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 5.4e+143], N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0 + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + N[(N[(-2.0 * N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.4000000000000003e143

   1. Initial program 90.8%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   7. Applied rewrites 95.7%

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

   if 5.4000000000000003e143 < x.im

   1. Initial program 59.1%

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

      1. lift--.f64 N/A

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

      2. 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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 89.9

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in x.re around 0

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

   9. Applied rewrites 89.9%

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

Alternative 7: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot x.im, -3, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-3 \cdot x.re\_m\right) \cdot x.im\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 7.5e+153)
    (* (fma (* x.im x.im) -3.0 (* x.re_m x.re_m)) x.re_m)
    (* (* (* -3.0 x.re_m) x.im) x.im))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 7.5e+153) {
		tmp = fma((x_46_im * x_46_im), -3.0, (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = ((-3.0 * x_46_re_m) * x_46_im) * x_46_im;
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 7.5e+153)
		tmp = Float64(fma(Float64(x_46_im * x_46_im), -3.0, Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = Float64(Float64(Float64(-3.0 * x_46_re_m) * x_46_im) * x_46_im);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 7.5e+153], N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0 + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.50000000000000065e153

   1. Initial program 91.1%

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

   3. 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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   7. Applied rewrites 95.8%

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

   if 7.50000000000000065e153 < x.im

   1. Initial program 54.5%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

         \[\leadsto \left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im \]

      6. metadata-eval N/A

         \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(-2 + -1\right)\right)\right) \cdot x.im \]

      7. distribute-rgt-out N/A

         \[\leadsto \left(x.im \cdot \left(-2 \cdot x.re + -1 \cdot x.re\right)\right) \cdot x.im \]

      8. lower-*.f64 N/A

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

      9. distribute-rgt-out N/A

         \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(-2 + -1\right)\right)\right) \cdot x.im \]

      10. metadata-eval N/A

          \[\leadsto \left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im \]

      12. *-commutative N/A

          \[\leadsto \left(\left(x.re \cdot -3\right) \cdot x.im\right) \cdot x.im \]

      13. lower-*.f64 88.8

          \[\leadsto \left(\left(x.re \cdot -3\right) \cdot x.im\right) \cdot x.im \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(x.re \cdot -3\right) \cdot x.im\right) \cdot x.im \]

      15. *-commutative N/A

          \[\leadsto \left(\left(-3 \cdot x.re\right) \cdot x.im\right) \cdot x.im \]

      16. lower-*.f64 88.8

          \[\leadsto \left(\left(-3 \cdot x.re\right) \cdot x.im\right) \cdot x.im \]

   7. Applied rewrites 88.8%

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

Alternative 8: 58.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

x.re\_m =     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)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((x_46re_m * x_46re_m) * x_46re_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

3. Taylor expanded in x.re around inf

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

   1. lower-pow.f64 52.3

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

5. Applied rewrites 52.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

      \[\leadsto {x.re}^{2} \cdot x.re \]

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 52.3

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

7. Applied rewrites 52.3%

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

Developer Target 1: 87.4% 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 2025064 
(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)))