math.cube on complex, imaginary part

Percentage Accurate: 82.5% → 99.7%

Time: 5.7s

Alternatives: 7

Speedup: 1.3×

• 44.4% 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: 82.5% 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: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;{\left(-x.im\_m\right)}^{3}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(x.im\_m \cdot \left(3 - \frac{x.im\_m}{x.re \cdot x.re} \cdot x.im\_m\right)\right) \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3 \cdot \frac{x.re}{x.im\_m}, \frac{x.re}{x.im\_m}, -1\right) \cdot {x.im\_m}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_0 0.0)
      (pow (- x.im_m) 3.0)
      (if (<= t_0 INFINITY)
        (*
         (* (* x.im_m (- 3.0 (* (/ x.im_m (* x.re x.re)) x.im_m))) x.re)
         x.re)
        (*
         (fma (* 3.0 (/ x.re x.im_m)) (/ x.re x.im_m) -1.0)
         (pow x.im_m 3.0)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = pow(-x_46_im_m, 3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((x_46_im_m * (3.0 - ((x_46_im_m / (x_46_re * x_46_re)) * x_46_im_m))) * x_46_re) * x_46_re;
	} else {
		tmp = fma((3.0 * (x_46_re / x_46_im_m)), (x_46_re / x_46_im_m), -1.0) * pow(x_46_im_m, 3.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(-x_46_im_m) ^ 3.0;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(x_46_im_m * Float64(3.0 - Float64(Float64(x_46_im_m / Float64(x_46_re * x_46_re)) * x_46_im_m))) * x_46_re) * x_46_re);
	else
		tmp = Float64(fma(Float64(3.0 * Float64(x_46_re / x_46_im_m)), Float64(x_46_re / x_46_im_m), -1.0) * (x_46_im_m ^ 3.0));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = 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$46$im$95$m), $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$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 0.0], N[Power[(-x$46$im$95$m), 3.0], $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x$46$im$95$m * N[(3.0 - N[(N[(x$46$im$95$m / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(N[(3.0 * N[(x$46$re / x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$re / x$46$im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 0.0

   1. Initial program 94.0%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. cube-neg-rev N/A

         \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

      4. lower-neg.f64 62.0

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

   5. Applied rewrites 62.0%

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

   if 0.0 < (+.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)) < +inf.0

   1. Initial program 92.7%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   7. Applied rewrites 67.8%

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

   9. Applied rewrites 86.3%

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

   if +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 0.0%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot \frac{x.re}{x.im}, \frac{x.re}{x.im}, -1\right) \cdot {x.im}^{3}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-320} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -2e-320) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* (* (* 3.0 x.im_m) x.re) x.re)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-320) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-320) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -2e-320) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -2e-320) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(Float64(3.0 * x_46_im_m) * x_46_re) * x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -2e-320) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = 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$46$im$95$m), $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$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -2e-320], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(N[(3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $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)) < -1.99998e-320 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 73.2%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. cube-neg-rev N/A

         \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

      4. lower-neg.f64 51.8

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

   5. Applied rewrites 51.8%

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

   7. Applied rewrites 51.8%

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

   if -1.99998e-320 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.5%

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 72.2%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -2 \cdot 10^{-320} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-320} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot x.re\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -2e-320) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* 3.0 (* (* x.im_m x.re) x.re))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-320) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = 3.0 * ((x_46_im_m * x_46_re) * x_46_re);
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-320) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = 3.0 * ((x_46_im_m * x_46_re) * x_46_re);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -2e-320) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = 3.0 * ((x_46_im_m * x_46_re) * x_46_re)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -2e-320) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_im_m * x_46_re) * x_46_re));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -2e-320) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = 3.0 * ((x_46_im_m * x_46_re) * x_46_re);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = 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$46$im$95$m), $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$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -2e-320], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(3.0 * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $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)) < -1.99998e-320 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 73.2%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. cube-neg-rev N/A

         \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

      4. lower-neg.f64 51.8

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

   5. Applied rewrites 51.8%

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

   7. Applied rewrites 51.8%

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

   if -1.99998e-320 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.5%

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 72.1%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -2 \cdot 10^{-320} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 3.1 \cdot 10^{+165}:\\ \;\;\;\;\left(-\mathsf{fma}\left(-3 \cdot x.re, x.re, x.im\_m \cdot x.im\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im\_m \cdot \left(3 - \frac{x.im\_m}{x.re \cdot x.re} \cdot x.im\_m\right)\right) \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 3.1e+165)
    (* (- (fma (* -3.0 x.re) x.re (* x.im_m x.im_m))) x.im_m)
    (* (* (* x.im_m (- 3.0 (* (/ x.im_m (* x.re x.re)) x.im_m))) x.re) x.re))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 3.1e+165) {
		tmp = -fma((-3.0 * x_46_re), x_46_re, (x_46_im_m * x_46_im_m)) * x_46_im_m;
	} else {
		tmp = ((x_46_im_m * (3.0 - ((x_46_im_m / (x_46_re * x_46_re)) * x_46_im_m))) * x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 3.1e+165)
		tmp = Float64(Float64(-fma(Float64(-3.0 * x_46_re), x_46_re, Float64(x_46_im_m * x_46_im_m))) * x_46_im_m);
	else
		tmp = Float64(Float64(Float64(x_46_im_m * Float64(3.0 - Float64(Float64(x_46_im_m / Float64(x_46_re * x_46_re)) * x_46_im_m))) * x_46_re) * x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 3.1e+165], N[((-N[(N[(-3.0 * x$46$re), $MachinePrecision] * x$46$re + N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]) * x$46$im$95$m), $MachinePrecision], N[(N[(N[(x$46$im$95$m * N[(3.0 - N[(N[(x$46$im$95$m / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.1000000000000002e165

   1. Initial program 89.1%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 95.5%

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

   if 3.1000000000000002e165 < x.re

   1. Initial program 60.8%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 60.8%

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

   9. Applied rewrites 94.6%

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

Alternative 5: 93.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 3.1 \cdot 10^{+165}:\\ \;\;\;\;\left(-\mathsf{fma}\left(-3 \cdot x.re, x.re, x.im\_m \cdot x.im\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 3.1e+165)
    (* (- (fma (* -3.0 x.re) x.re (* x.im_m x.im_m))) x.im_m)
    (* (* (* 3.0 x.im_m) x.re) x.re))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 3.1e+165) {
		tmp = -fma((-3.0 * x_46_re), x_46_re, (x_46_im_m * x_46_im_m)) * x_46_im_m;
	} else {
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 3.1e+165)
		tmp = Float64(Float64(-fma(Float64(-3.0 * x_46_re), x_46_re, Float64(x_46_im_m * x_46_im_m))) * x_46_im_m);
	else
		tmp = Float64(Float64(Float64(3.0 * x_46_im_m) * x_46_re) * x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 3.1e+165], N[((-N[(N[(-3.0 * x$46$re), $MachinePrecision] * x$46$re + N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]) * x$46$im$95$m), $MachinePrecision], N[(N[(N[(3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.1000000000000002e165

   1. Initial program 89.1%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 95.5%

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

   if 3.1000000000000002e165 < x.re

   1. Initial program 60.8%

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   7. Applied rewrites 94.6%

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

Alternative 6: 60.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 4e+165)
    (* (* (- x.im_m) x.im_m) x.im_m)
    (* (* x.im_m x.im_m) x.im_m))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 4e+165) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Fortran

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

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

real(8) function code(x_46im_s, x_46re, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 4d+165) then
        tmp = (-x_46im_m * x_46im_m) * x_46im_m
    else
        tmp = (x_46im_m * x_46im_m) * x_46im_m
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 4e+165) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 4e+165:
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = (x_46_im_m * x_46_im_m) * x_46_im_m
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 4e+165)
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 4e+165)
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = (x_46_im_m * x_46_im_m) * x_46_im_m;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 4e+165], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.9999999999999996e165

   1. Initial program 89.1%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. cube-neg-rev N/A

         \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

      4. lower-neg.f64 61.2

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

   5. Applied rewrites 61.2%

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

   7. Applied rewrites 61.1%

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

   if 3.9999999999999996e165 < x.re

   1. Initial program 60.8%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. cube-neg-rev N/A

         \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

      4. lower-neg.f64 6.5

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

   5. Applied rewrites 6.5%

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

   7. Applied rewrites 26.0%

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

Alternative 7: 21.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * ((x_46_im_m * x_46_im_m) * x_46_im_m);
}

Fortran

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

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

real(8) function code(x_46im_s, x_46re, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * ((x_46im_m * x_46im_m) * x_46im_m)
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * ((x_46_im_m * x_46_im_m) * x_46_im_m);
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	return x_46_im_s * ((x_46_im_m * x_46_im_m) * x_46_im_m)

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m))
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * ((x_46_im_m * x_46_im_m) * x_46_im_m);
end

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

   2. cube-neg-rev N/A

      \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(x.im\right)\right)}^{3}} \]

   4. lower-neg.f64 53.3

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

5. Applied rewrites 53.3%

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

7. Applied rewrites 23.3%

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

Developer Target 1: 91.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025007 
(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)))