math.cube on complex, imaginary part

Percentage Accurate: 82.4% → 96.4%

Time: 2.3s

Alternatives: 6

Speedup: 1.3×

• 44.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.4% 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: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.6 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(3, x.re\_m \cdot x.re\_m, \left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot 3\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 1.6e+152)
   (* (fma 3.0 (* x.re_m x.re_m) (* (- x.im) x.im)) x.im)
   (* (* (* x.im x.re_m) 3.0) 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 <= 1.6e+152) {
		tmp = fma(3.0, (x_46_re_m * x_46_re_m), (-x_46_im * x_46_im)) * x_46_im;
	} else {
		tmp = ((x_46_im * x_46_re_m) * 3.0) * 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 <= 1.6e+152)
		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(x_46_im * x_46_re_m) * 3.0) * 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, 1.6e+152], 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[(x$46$im * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.60000000000000003e152

   1. Initial program 81.1%

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

   3. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{x.im \cdot \left(-1 \cdot {x.im}^{2} + \left(2 \cdot {x.re}^{2} + {x.re}^{2}\right)\right)} \]
   4. 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 88.6

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

   5. Applied rewrites 88.6%

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

   if 1.60000000000000003e152 < x.re

   1. Initial program 56.6%

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

   3. Taylor expanded in x.re around inf

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

      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 72.7

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

   5. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\left(3 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right)} \]
   6. 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. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 96.6

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

   7. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot \color{blue}{x.re} \]
   8. 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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 96.7

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

   9. Applied rewrites 96.7%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.6 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(3, x.re \cdot x.re, \left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot 3\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-319} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot 3\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* 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 (or (<= t_0 -5e-319) (not (<= t_0 INFINITY)))
     (* (* (- x.im) x.im) x.im)
     (* (* (* x.im x.re_m) 3.0) x.re_m))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (((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_0 <= -5e-319) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = ((x_46_im * x_46_re_m) * 3.0) * x_46_re_m;
	}
	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_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_0 <= -5e-319) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = ((x_46_im * x_46_re_m) * 3.0) * x_46_re_m;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (((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_0 <= -5e-319) or not (t_0 <= math.inf):
		tmp = (-x_46_im * x_46_im) * x_46_im
	else:
		tmp = ((x_46_im * x_46_re_m) * 3.0) * x_46_re_m
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = 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_0 <= -5e-319) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	else
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * 3.0) * x_46_re_m);
	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_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_0 <= -5e-319) || ~((t_0 <= Inf)))
		tmp = (-x_46_im * x_46_im) * x_46_im;
	else
		tmp = ((x_46_im * x_46_re_m) * 3.0) * x_46_re_m;
	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[(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[Or[LessEqual[t$95$0, -5e-319], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.9999937e-319 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 66.4%

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

   3. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{x.im \cdot \left(-1 \cdot {x.im}^{2} + \left(2 \cdot {x.re}^{2} + {x.re}^{2}\right)\right)} \]
   4. 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 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in x.re around 0

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

      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. lower-*.f64 N/A

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

      5. lower-neg.f64 50.5

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

   8. Applied rewrites 50.5%

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

   if -4.9999937e-319 < (+.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 92.6%

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

   3. Taylor expanded in x.re around inf

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

      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 59.0

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

   5. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\left(3 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right)} \]
   6. 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. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 66.1

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

   7. Applied rewrites 66.1%

      \[\leadsto \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot \color{blue}{x.re} \]
   8. 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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.2

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

   9. Applied rewrites 66.2%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-319} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot 3\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-319} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot \left(x.im \cdot x.re\_m\right)\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* 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 (or (<= t_0 -5e-319) (not (<= t_0 INFINITY)))
     (* (* (- x.im) x.im) x.im)
     (* (* x.re_m (* x.im x.re_m)) 3.0))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (((x_46_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_0 <= -5e-319) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_re_m * (x_46_im * x_46_re_m)) * 3.0;
	}
	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_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_0 <= -5e-319) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_re_m * (x_46_im * x_46_re_m)) * 3.0;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (((x_46_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_0 <= -5e-319) or not (t_0 <= math.inf):
		tmp = (-x_46_im * x_46_im) * x_46_im
	else:
		tmp = (x_46_re_m * (x_46_im * x_46_re_m)) * 3.0
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(Float64(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_0 <= -5e-319) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	else
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_im * x_46_re_m)) * 3.0);
	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_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_0 <= -5e-319) || ~((t_0 <= Inf)))
		tmp = (-x_46_im * x_46_im) * x_46_im;
	else
		tmp = (x_46_re_m * (x_46_im * x_46_re_m)) * 3.0;
	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[(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[Or[LessEqual[t$95$0, -5e-319], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(x$46$re$95$m * N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.9999937e-319 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 66.4%

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

   3. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{x.im \cdot \left(-1 \cdot {x.im}^{2} + \left(2 \cdot {x.re}^{2} + {x.re}^{2}\right)\right)} \]
   4. 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 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in x.re around 0

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

      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. lower-*.f64 N/A

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

      5. lower-neg.f64 50.5

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

   8. Applied rewrites 50.5%

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

   if -4.9999937e-319 < (+.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 92.6%

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

   3. Taylor expanded in x.re around inf

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

      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 59.0

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

   5. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\left(3 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right)} \]
   6. 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. pow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 59.0

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

   7. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 66.2

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

   9. Applied rewrites 66.2%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-319} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-319} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\_m\right) \cdot \left(3 \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* 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 (or (<= t_0 -5e-319) (not (<= t_0 INFINITY)))
     (* (* (- x.im) x.im) x.im)
     (* (* x.im x.re_m) (* 3.0 x.re_m)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (((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_0 <= -5e-319) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_im * x_46_re_m) * (3.0 * x_46_re_m);
	}
	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_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_0 <= -5e-319) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_im * x_46_re_m) * (3.0 * x_46_re_m);
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (((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_0 <= -5e-319) or not (t_0 <= math.inf):
		tmp = (-x_46_im * x_46_im) * x_46_im
	else:
		tmp = (x_46_im * x_46_re_m) * (3.0 * x_46_re_m)
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = 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_0 <= -5e-319) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	else
		tmp = Float64(Float64(x_46_im * x_46_re_m) * Float64(3.0 * x_46_re_m));
	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_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_0 <= -5e-319) || ~((t_0 <= Inf)))
		tmp = (-x_46_im * x_46_im) * x_46_im;
	else
		tmp = (x_46_im * x_46_re_m) * (3.0 * x_46_re_m);
	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[(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[Or[LessEqual[t$95$0, -5e-319], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * N[(3.0 * 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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.9999937e-319 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 66.4%

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

   3. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{x.im \cdot \left(-1 \cdot {x.im}^{2} + \left(2 \cdot {x.re}^{2} + {x.re}^{2}\right)\right)} \]
   4. 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 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in x.re around 0

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

      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. lower-*.f64 N/A

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

      5. lower-neg.f64 50.5

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

   8. Applied rewrites 50.5%

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

   if -4.9999937e-319 < (+.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 92.6%

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

   3. Taylor expanded in x.re around inf

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

      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 59.0

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

   5. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\left(3 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right)} \]
   6. 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. pow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 59.0

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

   7. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 66.2

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

   9. Applied rewrites 66.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 66.2

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

   11. Applied rewrites 66.2%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-319} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(3 \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-319} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.im\right) \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* 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 (or (<= t_0 -5e-319) (not (<= t_0 INFINITY)))
     (* (* (- 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 t_0 = (((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_0 <= -5e-319) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-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;
}

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_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_0 <= -5e-319) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-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;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (((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_0 <= -5e-319) or not (t_0 <= math.inf):
		tmp = (-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)
	t_0 = 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_0 <= -5e-319) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	else
		tmp = Float64(Float64(3.0 * x_46_im) * Float64(x_46_re_m * x_46_re_m));
	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_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_0 <= -5e-319) || ~((t_0 <= Inf)))
		tmp = (-x_46_im * x_46_im) * x_46_im;
	else
		tmp = (3.0 * x_46_im) * (x_46_re_m * x_46_re_m);
	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[(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[Or[LessEqual[t$95$0, -5e-319], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(3.0 * x$46$im), $MachinePrecision] * N[(x$46$re$95$m * 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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.9999937e-319 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 66.4%

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

   3. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{x.im \cdot \left(-1 \cdot {x.im}^{2} + \left(2 \cdot {x.re}^{2} + {x.re}^{2}\right)\right)} \]
   4. 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 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in x.re around 0

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

      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. lower-*.f64 N/A

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

      5. lower-neg.f64 50.5

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

   8. Applied rewrites 50.5%

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

   if -4.9999937e-319 < (+.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 92.6%

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

   3. Taylor expanded in x.re around inf

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

      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 59.0

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

   5. Applied rewrites 59.0%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-319} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. 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 78.1%

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

3. Taylor expanded in x.im around 0

   \[\leadsto \color{blue}{x.im \cdot \left(-1 \cdot {x.im}^{2} + \left(2 \cdot {x.re}^{2} + {x.re}^{2}\right)\right)} \]
4. 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 84.7

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

5. Applied rewrites 84.7%

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

6. Taylor expanded in x.re around 0

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

   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. lower-*.f64 N/A

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

   5. lower-neg.f64 54.8

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

8. Applied rewrites 54.8%

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

Reproduce

herbie shell --seed 2025085 
(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)))