math.cube on complex, real part

Percentage Accurate: 82.3% → 99.8%

Time: 11.1s

Alternatives: 8

Speedup: 1.2×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 82.3% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 2 \cdot 10^{+234}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot \left(x.im + x.im\right), -x.im, \left(x.re\_m \cdot \left(x.re\_m + x.im\right)\right) \cdot \left(x.re\_m - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m - x.im\right)\right)\\ \end{array} \end{array} \]

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.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       2e+234)
    (fma
     (* x.re_m (+ x.im x.im))
     (- x.im)
     (* (* x.re_m (+ x.re_m x.im)) (- x.re_m x.im)))
    (* x.re_m (* (+ x.re_m x.im) (- x.re_m 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 2e+234) {
		tmp = fma((x_46_re_m * (x_46_im + x_46_im)), -x_46_im, ((x_46_re_m * (x_46_re_m + x_46_im)) * (x_46_re_m - x_46_im)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - 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(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 2e+234)
		tmp = fma(Float64(x_46_re_m * Float64(x_46_im + x_46_im)), Float64(-x_46_im), Float64(Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)) * Float64(x_46_re_m - x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m - 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[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+234], N[(N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision] * (-x$46$im) + N[(N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $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)) < 2.00000000000000004e234

   1. Initial program 94.9%

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

   4. Applied rewrites 99.7%

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

   if 2.00000000000000004e234 < (-.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 54.4%

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

   4. Applied rewrites 74.4%

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

   6. Applied rewrites 88.6%

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

   7. Taylor expanded in x.re around inf

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

   9. Applied rewrites 87.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{+234}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -5 \cdot 10^{-304}:\\ \;\;\;\;x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m - x.im\right)\right)\\ \end{array} \end{array} \]

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.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       -5e-304)
    (* x.im (* (* x.re_m x.im) -3.0))
    (* x.re_m (* (+ x.re_m x.im) (- x.re_m 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304) {
		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_im) * (x_46_re_m - 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_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-5d-304)) then
        tmp = x_46im * ((x_46re_m * x_46im) * (-3.0d0))
    else
        tmp = x_46re_m * ((x_46re_m + x_46im) * (x_46re_m - 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304) {
		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_im) * (x_46_re_m - 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304:
		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_im) * (x_46_re_m - 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(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -5e-304)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * -3.0));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m - 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304)
		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_im) * (x_46_re_m - 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[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-304], N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $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)) < -4.99999999999999965e-304

   1. Initial program 90.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.re around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 54.5

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

   7. Applied rewrites 54.5%

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

   if -4.99999999999999965e-304 < (-.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 76.9%

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 57.8%

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

   7. Taylor expanded in x.re around inf

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

   9. Applied rewrites 80.7%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-304}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -5 \cdot 10^{-304}:\\ \;\;\;\;x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       -5e-304)
    (* 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304) {
		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_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-5d-304)) 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304) {
		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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304:
		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(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -5e-304)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * -3.0));
	else
		tmp = Float64(x_46_re_m * Float64(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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304)
		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[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-304], N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $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)) < -4.99999999999999965e-304

   1. Initial program 90.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.re around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 54.5

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

   7. Applied rewrites 54.5%

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

   if -4.99999999999999965e-304 < (-.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 76.9%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

         \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x.re \cdot {x.re}^{2}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-304}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -5 \cdot 10^{-304}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       -5e-304)
    (* 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304) {
		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_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-5d-304)) 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304) {
		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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304:
		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(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -5e-304)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * -3.0)));
	else
		tmp = Float64(x_46_re_m * Float64(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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304)
		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[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-304], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $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)) < -4.99999999999999965e-304

   1. Initial program 90.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.re around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 54.5

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

   7. Applied rewrites 54.5%

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

      1. associate-*l* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 54.5

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

   9. Applied rewrites 54.5%

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

   if -4.99999999999999965e-304 < (-.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 76.9%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

         \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x.re \cdot {x.re}^{2}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-304}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -5 \cdot 10^{-304}:\\ \;\;\;\;-x.re\_m \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       -5e-304)
    (- (* x.re_m (* x.im x.im)))
    (* x.re_m (* x.re_m x.re_m)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304) {
		tmp = -(x_46_re_m * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

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_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-5d-304)) then
        tmp = -(x_46re_m * (x_46im * x_46im))
    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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304) {
		tmp = -(x_46_re_m * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304:
		tmp = -(x_46_re_m * (x_46_im * x_46_im))
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -5e-304)
		tmp = Float64(-Float64(x_46_re_m * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-304)
		tmp = -(x_46_re_m * (x_46_im * x_46_im));
	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[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-304], (-N[(x$46$re$95$m * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $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)) < -4.99999999999999965e-304

   1. Initial program 90.7%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 25.4

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

   9. Applied rewrites 25.4%

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

   if -4.99999999999999965e-304 < (-.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 76.9%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

         \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x.re \cdot {x.re}^{2}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-304}:\\ \;\;\;\;-x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re\_m \cdot \left(x.im \cdot -3\right), x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m - x.im\right)\right)\\ \end{array} \end{array} \]

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 2e+78)
    (fma x.im (* x.re_m (* x.im -3.0)) (* x.re_m (* x.re_m x.re_m)))
    (* x.re_m (* (+ x.re_m x.im) (- x.re_m 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 <= 2e+78) {
		tmp = fma(x_46_im, (x_46_re_m * (x_46_im * -3.0)), (x_46_re_m * (x_46_re_m * x_46_re_m)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - 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 (x_46_re_m <= 2e+78)
		tmp = fma(x_46_im, Float64(x_46_re_m * Float64(x_46_im * -3.0)), Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m - 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[x$46$re$95$m, 2e+78], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.00000000000000002e78

   1. Initial program 84.0%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 87.3

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

   5. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 90.1

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

   7. Applied rewrites 90.1%

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

   if 2.00000000000000002e78 < x.re

   1. Initial program 72.9%

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

   4. Applied rewrites 87.5%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x.re around inf

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

   9. Applied rewrites 100.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re \cdot \left(x.im \cdot -3\right), x.re \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.2% accurate, 3.6× speedup

Math

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

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (* x.re_s (* x.re_m (* 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(x_46_re_m * Float64(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[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Taylor expanded in x.re around inf

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

   1. cube-mult N/A

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

   2. unpow2 N/A

      \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{x.re \cdot {x.re}^{2}} \]

   4. unpow2 N/A

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

   5. lower-*.f64 60.0

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

5. Applied rewrites 60.0%

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

Alternative 8: 3.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.im \cdot 2\right) \end{array} \]

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.im 2.0)))

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) {
	return x_46_re_s * (x_46_im * 2.0);
}

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_46im * 2.0d0)
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_im * 2.0);
}

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_im * 2.0)

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(x_46_im * 2.0))
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_im * 2.0);
end

Wolfram

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

Derivation

1. Initial program 81.9%

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

4. Applied rewrites 91.6%

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

6. Applied rewrites 60.3%

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

7. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{2 \cdot x.im} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{x.im \cdot 2} \]

   2. lower-*.f64 3.4

      \[\leadsto \color{blue}{x.im \cdot 2} \]

9. Applied rewrites 3.4%

   \[\leadsto \color{blue}{x.im \cdot 2} \]
10. Add Preprocessing

Developer Target 1: 87.0% 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 2024219 
(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)))