math.cube on complex, real part

Percentage Accurate: 82.4% → 99.8%

Time: 5.6s

Alternatives: 6

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5.8 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im\_m \cdot x.im\_m, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{\frac{x.re}{x.im\_m}}{x.im\_m}, x.re, -3\right) \cdot x.re\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 5.8e+106)
   (* (fma -3.0 (* x.im_m x.im_m) (* x.re x.re)) x.re)
   (* (* (* (fma (/ (/ x.re x.im_m) x.im_m) x.re -3.0) x.re) x.im_m) x.im_m)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5.8e+106) {
		tmp = fma(-3.0, (x_46_im_m * x_46_im_m), (x_46_re * x_46_re)) * x_46_re;
	} else {
		tmp = ((fma(((x_46_re / x_46_im_m) / x_46_im_m), x_46_re, -3.0) * x_46_re) * x_46_im_m) * x_46_im_m;
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5.8e+106)
		tmp = Float64(fma(-3.0, Float64(x_46_im_m * x_46_im_m), Float64(x_46_re * x_46_re)) * x_46_re);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(x_46_re / x_46_im_m) / x_46_im_m), x_46_re, -3.0) * x_46_re) * x_46_im_m) * x_46_im_m);
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 5.8e+106], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(N[(N[(N[(N[(x$46$re / x$46$im$95$m), $MachinePrecision] / x$46$im$95$m), $MachinePrecision] * x$46$re + -3.0), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.8000000000000004e106

   1. Initial program 88.8%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft1-in N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if 5.8000000000000004e106 < x.im

   1. Initial program 59.9%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 99.9%

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

Alternative 2: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.im\_m \leq -4 \cdot 10^{-314}:\\ \;\;\;\;\left(\left(-3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.im_m))
      -4e-314)
   (* (* (* -3.0 x.im_m) x.re) x.im_m)
   (* (* x.re x.re) x.re)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_im_m)) <= -4e-314) {
		tmp = ((-3.0 * x_46_im_m) * x_46_re) * x_46_im_m;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

x.im_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, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46im_m * x_46re)) * x_46im_m)) <= (-4d-314)) then
        tmp = (((-3.0d0) * x_46im_m) * x_46re) * x_46im_m
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_im_m)) <= -4e-314) {
		tmp = ((-3.0 * x_46_im_m) * x_46_re) * x_46_im_m;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_im_m)) <= -4e-314:
		tmp = ((-3.0 * x_46_im_m) * x_46_re) * x_46_im_m
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_im_m)) <= -4e-314)
		tmp = Float64(Float64(Float64(-3.0 * x_46_im_m) * x_46_re) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_im_m)) <= -4e-314)
		tmp = ((-3.0 * x_46_im_m) * x_46_re) * x_46_im_m;
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-314], N[(N[(N[(-3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -3.9999999999e-314

   1. Initial program 93.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

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

   7. Applied rewrites 61.6%

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

   if -3.9999999999e-314 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 80.1%

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

   3. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 76.1

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 76.0%

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

Alternative 3: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.im\_m \leq -4 \cdot 10^{-314}:\\ \;\;\;\;-3 \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.im_m))
      -4e-314)
   (* -3.0 (* (* x.im_m x.re) x.im_m))
   (* (* x.re x.re) x.re)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_im_m)) <= -4e-314) {
		tmp = -3.0 * ((x_46_im_m * x_46_re) * x_46_im_m);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

x.im_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, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46im_m * x_46re)) * x_46im_m)) <= (-4d-314)) then
        tmp = (-3.0d0) * ((x_46im_m * x_46re) * x_46im_m)
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_im_m)) <= -4e-314) {
		tmp = -3.0 * ((x_46_im_m * x_46_re) * x_46_im_m);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_im_m)) <= -4e-314:
		tmp = -3.0 * ((x_46_im_m * x_46_re) * x_46_im_m)
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_im_m)) <= -4e-314)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im_m * x_46_re) * x_46_im_m));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_im_m)) <= -4e-314)
		tmp = -3.0 * ((x_46_im_m * x_46_re) * x_46_im_m);
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-314], N[(-3.0 * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -3.9999999999e-314

   1. Initial program 93.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

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

   if -3.9999999999e-314 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 80.1%

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

   3. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 76.1

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 76.0%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.46 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im\_m \cdot x.im\_m, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x.re}{x.im\_m}, x.re, -3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 1.46e-30)
   (* (fma -3.0 (* x.im_m x.im_m) (* x.re x.re)) x.re)
   (* (* (fma (/ x.re x.im_m) x.re (* -3.0 x.im_m)) x.re) x.im_m)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.46e-30) {
		tmp = fma(-3.0, (x_46_im_m * x_46_im_m), (x_46_re * x_46_re)) * x_46_re;
	} else {
		tmp = (fma((x_46_re / x_46_im_m), x_46_re, (-3.0 * x_46_im_m)) * x_46_re) * x_46_im_m;
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.46e-30)
		tmp = Float64(fma(-3.0, Float64(x_46_im_m * x_46_im_m), Float64(x_46_re * x_46_re)) * x_46_re);
	else
		tmp = Float64(Float64(fma(Float64(x_46_re / x_46_im_m), x_46_re, Float64(-3.0 * x_46_im_m)) * x_46_re) * x_46_im_m);
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 1.46e-30], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(N[(N[(x$46$re / x$46$im$95$m), $MachinePrecision] * x$46$re + N[(-3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.4600000000000001e-30

   1. Initial program 88.0%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft1-in N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 92.8

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

   5. Applied rewrites 92.8%

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

   if 1.4600000000000001e-30 < x.im

   1. Initial program 76.2%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.2%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x.re around 0

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

   10. Applied rewrites 95.1%

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

   12. Applied rewrites 99.8%

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

Alternative 5: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im\_m \cdot x.im\_m, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-3, x.im\_m, \frac{x.re \cdot x.re}{x.im\_m}\right) \cdot x.re\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 5e+80)
   (* (fma -3.0 (* x.im_m x.im_m) (* x.re x.re)) x.re)
   (* (* (fma -3.0 x.im_m (/ (* x.re x.re) x.im_m)) x.re) x.im_m)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+80) {
		tmp = fma(-3.0, (x_46_im_m * x_46_im_m), (x_46_re * x_46_re)) * x_46_re;
	} else {
		tmp = (fma(-3.0, x_46_im_m, ((x_46_re * x_46_re) / x_46_im_m)) * x_46_re) * x_46_im_m;
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+80)
		tmp = Float64(fma(-3.0, Float64(x_46_im_m * x_46_im_m), Float64(x_46_re * x_46_re)) * x_46_re);
	else
		tmp = Float64(Float64(fma(-3.0, x_46_im_m, Float64(Float64(x_46_re * x_46_re) / x_46_im_m)) * x_46_re) * x_46_im_m);
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 5e+80], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(N[(-3.0 * x$46$im$95$m + N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.99999999999999961e80

   1. Initial program 88.6%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft1-in N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 93.7

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

   5. Applied rewrites 93.7%

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

   if 4.99999999999999961e80 < x.im

   1. Initial program 64.0%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x.re around 0

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

   10. Applied rewrites 91.9%

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

Alternative 6: 58.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	return (x_46_re * x_46_re) * x_46_re;
}

Fortran

x.im_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, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = (x_46re * x_46re) * x_46re
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	return (x_46_re * x_46_re) * x_46_re;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	return (x_46_re * x_46_re) * x_46_re

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	return Float64(Float64(x_46_re * x_46_re) * x_46_re)
end

MATLAB

x.im_m = abs(x_46_im);
function tmp = code(x_46_re, x_46_im_m)
	tmp = (x_46_re * x_46_re) * x_46_re;
end

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Taylor expanded in x.re around inf

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

   1. lower-pow.f64 63.5

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

5. Applied rewrites 63.5%

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

7. Applied rewrites 63.5%

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

Developer Target 1: 86.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025015 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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