math.cube on complex, real part

Percentage Accurate: 82.6% → 99.7%

Time: 9.2s

Alternatives: 8

Speedup: 1.4×

• 45.1% 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

real(8) function code(x_46re, x_46im)
    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: 82.6% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \frac{1}{x.re\_m - x.im}\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.re\_m}{t\_0} \cdot \left(x.im + x.re\_m\right) - \left(\left(x.im + x.im\right) \cdot x.re\_m\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x.im + x.re\_m\right) \cdot x.re\_m}{t\_0} - \left(x.im + x.im\right)\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_0 (/ 1.0 (- x.re_m x.im))))
   (*
    x.re_s
    (if (<=
         (-
          (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
          (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
         2e+51)
      (- (* (/ x.re_m t_0) (+ x.im x.re_m)) (* (* (+ x.im x.im) x.re_m) x.im))
      (- (/ (* (+ x.im x.re_m) x.re_m) t_0) (+ x.im x.im))))))

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 t_0 = 1.0 / (x_46_re_m - 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_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 2e+51) {
		tmp = ((x_46_re_m / t_0) * (x_46_im + x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im);
	} else {
		tmp = (((x_46_im + x_46_re_m) * x_46_re_m) / t_0) - (x_46_im + x_46_im);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (x_46re_m - x_46im)
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46im * x_46re_m) + (x_46im * x_46re_m)) * x_46im)) <= 2d+51) then
        tmp = ((x_46re_m / t_0) * (x_46im + x_46re_m)) - (((x_46im + x_46im) * x_46re_m) * x_46im)
    else
        tmp = (((x_46im + x_46re_m) * x_46re_m) / t_0) - (x_46im + x_46im)
    end if
    code = x_46re_s * tmp
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) {
	double t_0 = 1.0 / (x_46_re_m - 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_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 2e+51) {
		tmp = ((x_46_re_m / t_0) * (x_46_im + x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im);
	} else {
		tmp = (((x_46_im + x_46_re_m) * x_46_re_m) / t_0) - (x_46_im + x_46_im);
	}
	return x_46_re_s * tmp;
}

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):
	t_0 = 1.0 / (x_46_re_m - x_46_im)
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 2e+51:
		tmp = ((x_46_re_m / t_0) * (x_46_im + x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im)
	else:
		tmp = (((x_46_im + x_46_re_m) * x_46_re_m) / t_0) - (x_46_im + x_46_im)
	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)
	t_0 = Float64(1.0 / Float64(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_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= 2e+51)
		tmp = Float64(Float64(Float64(x_46_re_m / t_0) * Float64(x_46_im + x_46_re_m)) - Float64(Float64(Float64(x_46_im + x_46_im) * x_46_re_m) * x_46_im));
	else
		tmp = Float64(Float64(Float64(Float64(x_46_im + x_46_re_m) * x_46_re_m) / t_0) - Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = 1.0 / (x_46_re_m - x_46_im);
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 2e+51)
		tmp = ((x_46_re_m / t_0) * (x_46_im + x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im);
	else
		tmp = (((x_46_im + x_46_re_m) * x_46_re_m) / t_0) - (x_46_im + x_46_im);
	end
	tmp_2 = 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_] := Block[{t$95$0 = N[(1.0 / N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]}, 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$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], 2e+51], N[(N[(N[(x$46$re$95$m / t$95$0), $MachinePrecision] * N[(x$46$im + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 2e51

   1. Initial program 95.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. 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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. /-rgt-identity N/A

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

      4. associate-/r/ N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if 2e51 < (-.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 68.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. 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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 80.7

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

   4. Applied rewrites 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. clear-num N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. flip-- N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 80.7

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

   6. Applied rewrites 80.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. +-inverses N/A

         \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - x.im \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      9. +-inverses N/A

         \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - x.im \cdot \frac{0}{\color{blue}{0}} \]

      10. associate-*r/ N/A

          \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - \color{blue}{\frac{x.im \cdot 0}{0}} \]

      11. +-inverses N/A

          \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - \frac{x.im \cdot \color{blue}{\left(x.im - x.im\right)}}{0} \]

      12. distribute-lft-out-- N/A

          \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{0} \]

      13. +-inverses N/A

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

      14. flip-+ N/A

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

      15. lift-+.f64 85.3

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

   8. Applied rewrites 85.3%

      \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - \color{blue}{\left(x.im + x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.re}{\frac{1}{x.re - x.im}} \cdot \left(x.im + x.re\right) - \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x.im + x.re\right) \cdot x.re}{\frac{1}{x.re - x.im}} - \left(x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(x.re\_m - x.im\right) \cdot x.re\_m\right) \cdot \left(x.im + x.re\_m\right) - \left(\left(x.im + x.im\right) \cdot x.re\_m\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x.im + x.re\_m\right) \cdot x.re\_m}{\frac{1}{x.re\_m - x.im}} - \left(x.im + x.im\right)\\ \end{array} \end{array} \]

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 x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       5e-7)
    (-
     (* (* (- x.re_m x.im) x.re_m) (+ x.im x.re_m))
     (* (* (+ x.im x.im) x.re_m) x.im))
    (- (/ (* (+ x.im x.re_m) x.re_m) (/ 1.0 (- x.re_m x.im))) (+ x.im x.im)))))

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_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 5e-7) {
		tmp = (((x_46_re_m - x_46_im) * x_46_re_m) * (x_46_im + x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im);
	} else {
		tmp = (((x_46_im + x_46_re_m) * x_46_re_m) / (1.0 / (x_46_re_m - x_46_im))) - (x_46_im + x_46_im);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46im * x_46re_m) + (x_46im * x_46re_m)) * x_46im)) <= 5d-7) then
        tmp = (((x_46re_m - x_46im) * x_46re_m) * (x_46im + x_46re_m)) - (((x_46im + x_46im) * x_46re_m) * x_46im)
    else
        tmp = (((x_46im + x_46re_m) * x_46re_m) / (1.0d0 / (x_46re_m - x_46im))) - (x_46im + x_46im)
    end if
    code = x_46re_s * tmp
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) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 5e-7) {
		tmp = (((x_46_re_m - x_46_im) * x_46_re_m) * (x_46_im + x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im);
	} else {
		tmp = (((x_46_im + x_46_re_m) * x_46_re_m) / (1.0 / (x_46_re_m - x_46_im))) - (x_46_im + x_46_im);
	}
	return x_46_re_s * tmp;
}

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):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 5e-7:
		tmp = (((x_46_re_m - x_46_im) * x_46_re_m) * (x_46_im + x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im)
	else:
		tmp = (((x_46_im + x_46_re_m) * x_46_re_m) / (1.0 / (x_46_re_m - x_46_im))) - (x_46_im + x_46_im)
	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_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= 5e-7)
		tmp = Float64(Float64(Float64(Float64(x_46_re_m - x_46_im) * x_46_re_m) * Float64(x_46_im + x_46_re_m)) - Float64(Float64(Float64(x_46_im + x_46_im) * x_46_re_m) * x_46_im));
	else
		tmp = Float64(Float64(Float64(Float64(x_46_im + x_46_re_m) * x_46_re_m) / Float64(1.0 / Float64(x_46_re_m - x_46_im))) - Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 5e-7)
		tmp = (((x_46_re_m - x_46_im) * x_46_re_m) * (x_46_im + x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im);
	else
		tmp = (((x_46_im + x_46_re_m) * x_46_re_m) / (1.0 / (x_46_re_m - x_46_im))) - (x_46_im + x_46_im);
	end
	tmp_2 = 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$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * N[(x$46$im + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] / N[(1.0 / N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 4.99999999999999977e-7

   1. Initial program 95.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. 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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if 4.99999999999999977e-7 < (-.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 69.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. 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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 81.7

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

   4. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. clear-num N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. flip-- N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 81.8

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

   6. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. +-inverses N/A

         \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - x.im \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      9. +-inverses N/A

         \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - x.im \cdot \frac{0}{\color{blue}{0}} \]

      10. associate-*r/ N/A

          \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - \color{blue}{\frac{x.im \cdot 0}{0}} \]

      11. +-inverses N/A

          \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - \frac{x.im \cdot \color{blue}{\left(x.im - x.im\right)}}{0} \]

      12. distribute-lft-out-- N/A

          \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{0} \]

      13. +-inverses N/A

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

      14. flip-+ N/A

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

      15. lift-+.f64 83.3

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

   8. Applied rewrites 83.3%

      \[\leadsto \frac{\left(x.re + x.im\right) \cdot x.re}{\frac{1}{x.re - x.im}} - \color{blue}{\left(x.im + x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x.im + x.re\right) \cdot x.re}{\frac{1}{x.re - x.im}} - \left(x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq 10^{+297}:\\ \;\;\;\;\left(\left(x.re\_m - x.im\right) \cdot x.re\_m\right) \cdot \left(x.im + x.re\_m\right) - \left(\left(x.im + x.im\right) \cdot x.re\_m\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re\_m, \mathsf{fma}\left(\frac{x.re\_m}{x.im}, x.re\_m, -x.re\_m\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \end{array} \]

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 x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       1e+297)
    (-
     (* (* (- x.re_m x.im) x.re_m) (+ x.im x.re_m))
     (* (* (+ x.im x.im) x.re_m) x.im))
    (fma
     (+ x.im x.re_m)
     (* (fma (/ x.re_m x.im) x.re_m (- x.re_m)) x.im)
     (* 2.0 x.im)))))

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_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 1e+297) {
		tmp = (((x_46_re_m - x_46_im) * x_46_re_m) * (x_46_im + x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im);
	} else {
		tmp = fma((x_46_im + x_46_re_m), (fma((x_46_re_m / x_46_im), x_46_re_m, -x_46_re_m) * x_46_im), (2.0 * x_46_im));
	}
	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_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= 1e+297)
		tmp = Float64(Float64(Float64(Float64(x_46_re_m - x_46_im) * x_46_re_m) * Float64(x_46_im + x_46_re_m)) - Float64(Float64(Float64(x_46_im + x_46_im) * x_46_re_m) * x_46_im));
	else
		tmp = fma(Float64(x_46_im + x_46_re_m), Float64(fma(Float64(x_46_re_m / x_46_im), x_46_re_m, Float64(-x_46_re_m)) * x_46_im), Float64(2.0 * x_46_im));
	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$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], 1e+297], N[(N[(N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * N[(x$46$im + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * N[(N[(N[(x$46$re$95$m / x$46$im), $MachinePrecision] * x$46$re$95$m + (-x$46$re$95$m)), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(2.0 * x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1e297

   1. Initial program 96.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. 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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if 1e297 < (-.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 60.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. 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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 76.2

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

   4. Applied rewrites 76.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 76.2

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

   6. Applied rewrites 76.2%

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

   7. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\frac{{x.re}^{2}}{x.im} + \color{blue}{\left(0 - x.re\right)}\right) \cdot x.im\right) - \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im \]

      5. associate-+r- N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\color{blue}{\left(\left(\frac{{x.re}^{2}}{x.im} + 0\right) - x.re\right)} \cdot x.im\right) - \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im \]

      6. mul0-lft N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(\frac{{x.re}^{2}}{x.im} + \color{blue}{0 \cdot \frac{x.re}{x.im}}\right) - x.re\right) \cdot x.im\right) - \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im \]

      7. associate-*r/ N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(\frac{{x.re}^{2}}{x.im} + \color{blue}{\frac{0 \cdot x.re}{x.im}}\right) - x.re\right) \cdot x.im\right) - \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im \]

      8. metadata-eval N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(\frac{{x.re}^{2}}{x.im} + \frac{\color{blue}{\left(0 \cdot 0\right)} \cdot x.re}{x.im}\right) - x.re\right) \cdot x.im\right) - \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im \]

      9. associate-*r* N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(\frac{{x.re}^{2}}{x.im} + \frac{\color{blue}{0 \cdot \left(0 \cdot x.re\right)}}{x.im}\right) - x.re\right) \cdot x.im\right) - \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im \]

      10. metadata-eval N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(\frac{{x.re}^{2}}{x.im} + \frac{0 \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot x.re\right)}{x.im}\right) - x.re\right) \cdot x.im\right) - \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im \]

      11. distribute-rgt1-in N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(\frac{{x.re}^{2}}{x.im} + \frac{0 \cdot \color{blue}{\left(x.re + -1 \cdot x.re\right)}}{x.im}\right) - x.re\right) \cdot x.im\right) - \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im \]

      12. mul0-lft N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(\frac{{x.re}^{2}}{x.im} + \frac{\color{blue}{\left(0 \cdot x.re\right)} \cdot \left(x.re + -1 \cdot x.re\right)}{x.im}\right) - x.re\right) \cdot x.im\right) - \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im \]

      13. metadata-eval N/A

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

      14. distribute-rgt1-in N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

   9. Applied rewrites 76.2%

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

   11. Applied rewrites 90.8%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 10^{+297}:\\ \;\;\;\;\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re, \mathsf{fma}\left(\frac{x.re}{x.im}, x.re, -x.re\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\left(\left(x.re\_m - x.im\right) \cdot x.re\_m\right) \cdot \left(x.im + x.re\_m\right) - \left(\left(x.im + x.im\right) \cdot x.re\_m\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m, -3 \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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 x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       -5e-118)
    (-
     (* (* (- x.re_m x.im) x.re_m) (+ x.im x.re_m))
     (* (* (+ x.im x.im) x.re_m) x.im))
    (* (fma x.re_m x.re_m (* -3.0 (* x.im x.im))) 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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -5e-118) {
		tmp = (((x_46_re_m - x_46_im) * x_46_re_m) * (x_46_im + x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im);
	} else {
		tmp = fma(x_46_re_m, x_46_re_m, (-3.0 * (x_46_im * x_46_im))) * 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 (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_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -5e-118)
		tmp = Float64(Float64(Float64(Float64(x_46_re_m - x_46_im) * x_46_re_m) * Float64(x_46_im + x_46_re_m)) - Float64(Float64(Float64(x_46_im + x_46_im) * x_46_re_m) * x_46_im));
	else
		tmp = Float64(fma(x_46_re_m, x_46_re_m, Float64(-3.0 * Float64(x_46_im * x_46_im))) * 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[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$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -5e-118], N[(N[(N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * N[(x$46$im + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -5.00000000000000015e-118

   1. Initial program 92.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. 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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -5.00000000000000015e-118 < (-.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 81.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. 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

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 36.9%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-*.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft1-in N/A

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

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

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

      13. +-commutative N/A

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

      14. associate--l- N/A

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

      15. unsub-neg N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

   10. Applied rewrites 93.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re, -3 \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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 x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       -5e-305)
    (* (* (* x.im x.re_m) x.im) -3.0)
    (* (* 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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -5e-305) {
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46im * x_46re_m) + (x_46im * x_46re_m)) * x_46im)) <= (-5d-305)) then
        tmp = ((x_46im * x_46re_m) * x_46im) * (-3.0d0)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
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) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -5e-305) {
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

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):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -5e-305:
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0
	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 (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_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -5e-305)
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * x_46_im) * -3.0);
	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

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -5e-305)
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0;
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = 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$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -5e-305], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * -3.0), $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 (-.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)) < -4.99999999999999985e-305

   1. Initial program 92.9%

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

   3. Taylor expanded in x.im around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 41.7

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

   5. Applied rewrites 41.7%

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

   7. Applied rewrites 48.5%

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

   if -4.99999999999999985e-305 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 80.4%

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

   3. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 63.3

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

   5. Applied rewrites 63.3%

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

   7. Applied rewrites 63.2%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\left(\left(x.im \cdot x.im\right) \cdot x.re\_m\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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 x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       -5e-305)
    (* (* (* x.im x.im) x.re_m) -3.0)
    (* (* 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 * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -5e-305) {
		tmp = ((x_46_im * x_46_im) * x_46_re_m) * -3.0;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46im * x_46re_m) + (x_46im * x_46re_m)) * x_46im)) <= (-5d-305)) then
        tmp = ((x_46im * x_46im) * x_46re_m) * (-3.0d0)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
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) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -5e-305) {
		tmp = ((x_46_im * x_46_im) * x_46_re_m) * -3.0;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

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):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -5e-305:
		tmp = ((x_46_im * x_46_im) * x_46_re_m) * -3.0
	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 (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_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -5e-305)
		tmp = Float64(Float64(Float64(x_46_im * x_46_im) * x_46_re_m) * -3.0);
	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

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -5e-305)
		tmp = ((x_46_im * x_46_im) * x_46_re_m) * -3.0;
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = 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$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -5e-305], N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * -3.0), $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 (-.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)) < -4.99999999999999985e-305

   1. Initial program 92.9%

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

   3. Taylor expanded in x.im around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 41.7

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

   5. Applied rewrites 41.7%

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

   if -4.99999999999999985e-305 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 80.4%

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

   3. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 63.3

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

   5. Applied rewrites 63.3%

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

   7. Applied rewrites 63.2%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 4.6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right) \cdot -3\\ \end{array} \end{array} \]

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.im 4.6e+151)
    (* (fma -3.0 (* x.im x.im) (* x.re_m x.re_m)) x.re_m)
    (* (* (* x.im x.re_m) x.im) -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_im <= 4.6e+151) {
		tmp = fma(-3.0, (x_46_im * x_46_im), (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -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 (x_46_im <= 4.6e+151)
		tmp = Float64(fma(-3.0, Float64(x_46_im * x_46_im), Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * x_46_im) * -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[x$46$im, 4.6e+151], N[(N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.6000000000000002e151

   1. Initial program 88.7%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 94.8%

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

   if 4.6000000000000002e151 < x.im

   1. Initial program 52.0%

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

   3. Taylor expanded in x.im around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.6

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

   5. Applied rewrites 63.6%

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

   7. Applied rewrites 92.2%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.7% accurate, 3.6× speedup

Math

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

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 = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    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 85.0%

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

3. Taylor expanded in x.im around 0

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

   1. lower-pow.f64 59.7

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

5. Applied rewrites 59.7%

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

7. Applied rewrites 59.6%

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

Developer Target 1: 86.6% 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

real(8) function code(x_46re, x_46im)
    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 2024244 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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