math.cube on complex, imaginary part

Percentage Accurate: 83.0% → 96.5%

Time: 2.9s

Alternatives: 6

Speedup: 1.3×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 7 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(3 \cdot x.re\_m, x.re\_m, \left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 7e+157)
   (* (fma (* 3.0 x.re_m) x.re_m (* (- x.im) x.im)) x.im)
   (* (* (* 3.0 x.im) x.re_m) x.re_m)))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 7e+157) {
		tmp = fma((3.0 * x_46_re_m), x_46_re_m, (-x_46_im * x_46_im)) * x_46_im;
	} else {
		tmp = ((3.0 * x_46_im) * x_46_re_m) * x_46_re_m;
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 7e+157)
		tmp = Float64(fma(Float64(3.0 * x_46_re_m), x_46_re_m, Float64(Float64(-x_46_im) * x_46_im)) * x_46_im);
	else
		tmp = Float64(Float64(Float64(3.0 * x_46_im) * x_46_re_m) * x_46_re_m);
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 7e+157], N[(N[(N[(3.0 * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m + N[((-x$46$im) * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(N[(3.0 * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.00000000000000004e157

   1. Initial program 92.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, {x.re}^{2}, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(3 \cdot x.re, x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(3 \cdot x.re, x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      10. mul-1-neg N/A

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

      11. pow2 N/A

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

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

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

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(3 \cdot x.re, x.re, \left(-1 \cdot x.im\right) \cdot x.im\right) \cdot x.im \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(3 \cdot x.re, x.re, \left(-1 \cdot x.im\right) \cdot x.im\right) \cdot x.im \]

      15. mul-1-neg N/A

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

      16. lower-neg.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if 7.00000000000000004e157 < x.re

   1. Initial program 53.1%

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

   2. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 65.7

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

   4. Applied rewrites 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

Alternative 2: 60.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ t_1 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(3 \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* (* (- x.im) x.im) x.im))
        (t_1
         (+
          (* (- (* x.re_m x.re_m) (* x.im x.im)) x.im)
          (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.re_m))))
   (if (<= t_1 -1e-285)
     t_0
     (if (<= t_1 INFINITY) (* (* x.re_m x.im) (* 3.0 x.re_m)) t_0))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (-x_46_im * x_46_im) * x_46_im;
	double t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -1e-285) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_46_re_m * x_46_im) * (3.0 * x_46_re_m);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (-x_46_im * x_46_im) * x_46_im;
	double t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -1e-285) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re_m * x_46_im) * (3.0 * x_46_re_m);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (-x_46_im * x_46_im) * x_46_im
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if t_1 <= -1e-285:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (x_46_re_m * x_46_im) * (3.0 * x_46_re_m)
	else:
		tmp = t_0
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im)
	t_1 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_1 <= -1e-285)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(3.0 * x_46_re_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (-x_46_im * x_46_im) * x_46_im;
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if (t_1 <= -1e-285)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (x_46_re_m * x_46_im) * (3.0 * x_46_re_m);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]}, Block[{t$95$1 = 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$46$im), $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$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-285], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(3.0 * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.00000000000000007e-285 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 70.8%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, {x.re}^{2}, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 55.6

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

   7. Applied rewrites 55.6%

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

   if -1.00000000000000007e-285 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.9%

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

   2. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 60.7

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

   4. Applied rewrites 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 65.6

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

   6. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 65.6

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

   8. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 65.6

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

   10. Applied rewrites 65.6%

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

Alternative 3: 58.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ t_1 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.re\_m \cdot x.re\_m\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* (* (- x.im) x.im) x.im))
        (t_1
         (+
          (* (- (* x.re_m x.re_m) (* x.im x.im)) x.im)
          (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.re_m))))
   (if (<= t_1 -1e-285)
     t_0
     (if (<= t_1 INFINITY) (* (* 3.0 (* x.re_m x.re_m)) x.im) t_0))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (-x_46_im * x_46_im) * x_46_im;
	double t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -1e-285) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (3.0 * (x_46_re_m * x_46_re_m)) * x_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (-x_46_im * x_46_im) * x_46_im;
	double t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -1e-285) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (3.0 * (x_46_re_m * x_46_re_m)) * x_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (-x_46_im * x_46_im) * x_46_im
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if t_1 <= -1e-285:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (3.0 * (x_46_re_m * x_46_re_m)) * x_46_im
	else:
		tmp = t_0
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im)
	t_1 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_1 <= -1e-285)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(3.0 * Float64(x_46_re_m * x_46_re_m)) * x_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (-x_46_im * x_46_im) * x_46_im;
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if (t_1 <= -1e-285)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (3.0 * (x_46_re_m * x_46_re_m)) * x_46_im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]}, Block[{t$95$1 = 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$46$im), $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$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-285], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(3.0 * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.00000000000000007e-285 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 70.8%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, {x.re}^{2}, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 55.6

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

   7. Applied rewrites 55.6%

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

   if -1.00000000000000007e-285 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.9%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 60.7

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

   4. Applied rewrites 60.7%

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

Alternative 4: 58.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ t_1 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.re\_m\right) \cdot x.re\_m\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* (* (- x.im) x.im) x.im))
        (t_1
         (+
          (* (- (* x.re_m x.re_m) (* x.im x.im)) x.im)
          (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.re_m))))
   (if (<= t_1 -1e-285)
     t_0
     (if (<= t_1 INFINITY) (* (* (* 3.0 x.re_m) x.re_m) x.im) t_0))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (-x_46_im * x_46_im) * x_46_im;
	double t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -1e-285) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((3.0 * x_46_re_m) * x_46_re_m) * x_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (-x_46_im * x_46_im) * x_46_im;
	double t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -1e-285) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((3.0 * x_46_re_m) * x_46_re_m) * x_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (-x_46_im * x_46_im) * x_46_im
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if t_1 <= -1e-285:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = ((3.0 * x_46_re_m) * x_46_re_m) * x_46_im
	else:
		tmp = t_0
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im)
	t_1 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_1 <= -1e-285)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(3.0 * x_46_re_m) * x_46_re_m) * x_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (-x_46_im * x_46_im) * x_46_im;
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if (t_1 <= -1e-285)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = ((3.0 * x_46_re_m) * x_46_re_m) * x_46_im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]}, Block[{t$95$1 = 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$46$im), $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$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-285], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(3.0 * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.00000000000000007e-285 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 70.8%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, {x.re}^{2}, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 55.6

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

   7. Applied rewrites 55.6%

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

   if -1.00000000000000007e-285 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.9%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, {x.re}^{2}, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(3 \cdot x.re, x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(3 \cdot x.re, x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      10. mul-1-neg N/A

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

      11. pow2 N/A

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

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

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

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(3 \cdot x.re, x.re, \left(-1 \cdot x.im\right) \cdot x.im\right) \cdot x.im \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(3 \cdot x.re, x.re, \left(-1 \cdot x.im\right) \cdot x.im\right) \cdot x.im \]

      15. mul-1-neg N/A

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

      16. lower-neg.f64 94.9

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

   6. Applied rewrites 94.9%

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

   7. Taylor expanded in x.re around inf

      \[\leadsto \left(3 \cdot {x.re}^{2}\right) \cdot x.im \]
   8. Step-by-step derivation

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 60.7

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

   9. Applied rewrites 60.7%

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

Alternative 5: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(3, x.re\_m \cdot x.re\_m, -x.im \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 5.2e+153)
   (* (fma 3.0 (* x.re_m x.re_m) (- (* x.im x.im))) x.im)
   (* (* (* 3.0 x.im) x.re_m) x.re_m)))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5.2e+153) {
		tmp = fma(3.0, (x_46_re_m * x_46_re_m), -(x_46_im * x_46_im)) * x_46_im;
	} else {
		tmp = ((3.0 * x_46_im) * x_46_re_m) * x_46_re_m;
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 5.2e+153)
		tmp = Float64(fma(3.0, Float64(x_46_re_m * x_46_re_m), Float64(-Float64(x_46_im * x_46_im))) * x_46_im);
	else
		tmp = Float64(Float64(Float64(3.0 * x_46_im) * x_46_re_m) * x_46_re_m);
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 5.2e+153], N[(N[(3.0 * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + (-N[(x$46$im * x$46$im), $MachinePrecision])), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(N[(3.0 * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.1999999999999998e153

   1. Initial program 92.8%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, {x.re}^{2}, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 5.1999999999999998e153 < x.re

   1. Initial program 52.7%

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

   2. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 65.0

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

   4. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 87.4

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

   6. Applied rewrites 87.4%

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

Alternative 6: 59.2% accurate, 3.1× speedup

Math

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

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im) :precision binary64 (* (* (- x.im) x.im) x.im))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	return (-x_46_im * x_46_im) * x_46_im;
}

Fortran

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

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	return (-x_46_im * x_46_im) * x_46_im;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	return (-x_46_im * x_46_im) * x_46_im

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	return Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im)
end

MATLAB

x.re_m = abs(x_46_re);
function tmp = code(x_46_re_m, x_46_im)
	tmp = (-x_46_im * x_46_im) * x_46_im;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

2. Taylor expanded in x.im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(3, {x.re}^{2}, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

   7. pow2 N/A

      \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(3, x.re \cdot x.re, -1 \cdot {x.im}^{2}\right) \cdot x.im \]

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 88.2

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

4. Applied rewrites 88.2%

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

5. Taylor expanded in x.re around 0

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

   1. mul-1-neg N/A

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

   2. pow2 N/A

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

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

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

   4. mul-1-neg N/A

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

   5. lower-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f64 59.2

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

7. Applied rewrites 59.2%

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

Developer Target 1: 91.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025093 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

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

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