math.cube on complex, real part

Percentage Accurate: 82.4% → 96.7%

Time: 2.7s

Alternatives: 7

Speedup: 1.8×

• 44.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.4% 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: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, \left(x.im + x.re\_m\right) \cdot x.re\_m, \left(-2 \cdot \left(x.im \cdot x.re\_m\right)\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       2e-276)
    (fma
     (- x.re_m x.im)
     (* (+ x.im x.re_m) x.re_m)
     (* (* -2.0 (* x.im x.re_m)) x.im))
    (pow x.re_m 3.0))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= 2e-276) {
		tmp = fma((x_46_re_m - x_46_im), ((x_46_im + x_46_re_m) * x_46_re_m), ((-2.0 * (x_46_im * x_46_re_m)) * x_46_im));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= 2e-276)
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(Float64(x_46_im + x_46_re_m) * x_46_re_m), Float64(Float64(-2.0 * Float64(x_46_im * x_46_re_m)) * x_46_im));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], 2e-276], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + N[(N[(-2.0 * N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 2e-276

   1. Initial program 82.4%

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

   3. Applied rewrites 91.7%

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

   if 2e-276 < (-.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 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 58.2

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

   4. Applied rewrites 58.2%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot x.re\_m, x.re\_m, \left(\left(-3 \cdot x.re\_m\right) \cdot x.im\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       2e-276)
    (fma (* x.re_m x.re_m) x.re_m (* (* (* -3.0 x.re_m) x.im) x.im))
    (pow x.re_m 3.0))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= 2e-276) {
		tmp = fma((x_46_re_m * x_46_re_m), x_46_re_m, (((-3.0 * x_46_re_m) * x_46_im) * x_46_im));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= 2e-276)
		tmp = fma(Float64(x_46_re_m * x_46_re_m), x_46_re_m, Float64(Float64(Float64(-3.0 * x_46_re_m) * x_46_im) * x_46_im));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], 2e-276], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m + N[(N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 2e-276

   1. Initial program 82.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-neg-in N/A

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

   3. Applied rewrites 90.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares N/A

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

      15. lift-*.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. associate--r- N/A

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

   5. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 88.5

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

   9. Applied rewrites 88.5%

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

   if 2e-276 < (-.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 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 58.2

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

   4. Applied rewrites 58.2%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot x.re\_m, x.re\_m, \left(-3 \cdot \left(x.im \cdot x.re\_m\right)\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       2e-276)
    (fma (* x.re_m x.re_m) x.re_m (* (* -3.0 (* x.im x.re_m)) x.im))
    (pow x.re_m 3.0))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= 2e-276) {
		tmp = fma((x_46_re_m * x_46_re_m), x_46_re_m, ((-3.0 * (x_46_im * x_46_re_m)) * x_46_im));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= 2e-276)
		tmp = fma(Float64(x_46_re_m * x_46_re_m), x_46_re_m, Float64(Float64(-3.0 * Float64(x_46_im * x_46_re_m)) * x_46_im));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], 2e-276], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m + N[(N[(-3.0 * N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 2e-276

   1. Initial program 82.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-neg-in N/A

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

   3. Applied rewrites 90.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares N/A

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

      15. lift-*.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. associate--r- N/A

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

   5. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 88.6%

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

   if 2e-276 < (-.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 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 58.2

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

   4. Applied rewrites 58.2%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.2 \cdot 10^{-119}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot x.re\_m, -3 \cdot x.im, \left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.im \cdot -3, x.im, x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1.2e-119)
    (fma (* x.im x.re_m) (* -3.0 x.im) (* (* x.re_m x.re_m) x.re_m))
    (* x.re_m (fma (* x.im -3.0) x.im (* x.re_m x.re_m))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.2e-119) {
		tmp = fma((x_46_im * x_46_re_m), (-3.0 * x_46_im), ((x_46_re_m * x_46_re_m) * x_46_re_m));
	} else {
		tmp = x_46_re_m * fma((x_46_im * -3.0), x_46_im, (x_46_re_m * x_46_re_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1.2e-119)
		tmp = fma(Float64(x_46_im * x_46_re_m), Float64(-3.0 * x_46_im), Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m));
	else
		tmp = Float64(x_46_re_m * fma(Float64(x_46_im * -3.0), x_46_im, Float64(x_46_re_m * x_46_re_m)));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.2e-119], N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * N[(-3.0 * x$46$im), $MachinePrecision] + N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$im * -3.0), $MachinePrecision] * x$46$im + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.20000000000000004e-119

   1. Initial program 82.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-neg-in N/A

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

   3. Applied rewrites 90.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares N/A

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

      15. lift-*.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. associate--r- N/A

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

   5. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-rgt-neg-out N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. add-flip N/A

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

      16. distribute-lft-in N/A

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

   7. Applied rewrites 87.8%

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

   if 1.20000000000000004e-119 < x.re

   1. Initial program 82.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-neg-in N/A

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

   3. Applied rewrites 90.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares N/A

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

      15. lift-*.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. associate--r- N/A

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

   5. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 90.5

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

   7. Applied rewrites 90.5%

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

Alternative 5: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 6 \cdot 10^{+228}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.re\_m, x.re\_m, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 6e+228)
    (* x.re_m (fma x.re_m x.re_m (* -3.0 (* x.im x.im))))
    (* (* x.re_m x.re_m) x.re_m))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 6e+228) {
		tmp = x_46_re_m * fma(x_46_re_m, x_46_re_m, (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 6e+228)
		tmp = Float64(x_46_re_m * fma(x_46_re_m, x_46_re_m, Float64(-3.0 * Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 6e+228], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.0000000000000002e228

   1. Initial program 82.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-neg-in N/A

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

   3. Applied rewrites 90.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift--.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-+l+ N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. count-2 N/A

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

      22. associate-*l* N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 N/A

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

   5. Applied rewrites 90.5%

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

   if 6.0000000000000002e228 < x.re

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 58.2

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

   4. Applied rewrites 58.2%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 58.2

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

   6. Applied rewrites 58.2%

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

Alternative 6: 90.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.re\_m \cdot \mathsf{fma}\left(x.im \cdot -3, x.im, x.re\_m \cdot x.re\_m\right)\right) \end{array} \]

FPCore

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

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_re_m * fma((x_46_im * -3.0), x_46_im, (x_46_re_m * x_46_re_m)));
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(x_46_re_m * fma(Float64(x_46_im * -3.0), x_46_im, Float64(x_46_re_m * x_46_re_m))))
end

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. distribute-rgt-neg-in N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. distribute-rgt-neg-in N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-rgt-neg-in N/A

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

3. Applied rewrites 90.5%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. add-flip N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift--.f64 N/A

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

   14. difference-of-squares N/A

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

   15. lift-*.f64 N/A

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

   16. sub-negate-rev N/A

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

   17. associate--r- N/A

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

5. Applied rewrites 90.5%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

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

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 90.5

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

7. Applied rewrites 90.5%

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

Alternative 7: 58.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_46re_s, x_46re_m, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((x_46re_m * x_46re_m) * x_46re_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.re around inf

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

   1. lower-pow.f64 58.2

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

4. Applied rewrites 58.2%

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

   1. lift-pow.f64 N/A

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

   2. pow3 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-*.f64 58.2

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

6. Applied rewrites 58.2%

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

Developer Target 1: 87.2% 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 2025159 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform c (+ (* (* 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)))