math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 96.6%

Time: 8.7s

Alternatives: 8

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + t\_0\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-179}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 3\right) - {x.im\_m}^{3}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0 + \left(x.re \cdot x.im\_m\right) \cdot \left(\left(1 - \frac{x.im\_m}{x.re}\right) \cdot \left(x.re + x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-{x.im\_m}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))
        (t_1 (+ (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m))) t_0)))
   (*
    x.im_s
    (if (<= t_1 2e-179)
      (- (* x.re (* (* x.re x.im_m) 3.0)) (pow x.im_m 3.0))
      (if (<= t_1 INFINITY)
        (+ t_0 (* (* x.re x.im_m) (* (- 1.0 (/ x.im_m x.re)) (+ x.re x.im_m))))
        (- (pow x.im_m 3.0)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0;
	double tmp;
	if (t_1 <= 2e-179) {
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 3.0)) - pow(x_46_im_m, 3.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)));
	} else {
		tmp = -pow(x_46_im_m, 3.0);
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0;
	double tmp;
	if (t_1 <= 2e-179) {
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 3.0)) - Math.pow(x_46_im_m, 3.0);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)));
	} else {
		tmp = -Math.pow(x_46_im_m, 3.0);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0
	tmp = 0
	if t_1 <= 2e-179:
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 3.0)) - math.pow(x_46_im_m, 3.0)
	elif t_1 <= math.inf:
		tmp = t_0 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)))
	else:
		tmp = -math.pow(x_46_im_m, 3.0)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + t_0)
	tmp = 0.0
	if (t_1 <= 2e-179)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 3.0)) - (x_46_im_m ^ 3.0));
	elseif (t_1 <= Inf)
		tmp = Float64(t_0 + Float64(Float64(x_46_re * x_46_im_m) * Float64(Float64(1.0 - Float64(x_46_im_m / x_46_re)) * Float64(x_46_re + x_46_im_m))));
	else
		tmp = Float64(-(x_46_im_m ^ 3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0;
	tmp = 0.0;
	if (t_1 <= 2e-179)
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 3.0)) - (x_46_im_m ^ 3.0);
	elseif (t_1 <= Inf)
		tmp = t_0 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)));
	else
		tmp = -(x_46_im_m ^ 3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, 2e-179], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$0 + N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(N[(1.0 - N[(x$46$im$95$m / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision])]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 2e-179

   1. Initial program 94.5%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in x.re around 0 96.4%

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

   if 2e-179 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 92.1%

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

      1. difference-of-squares 92.1%

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

      2. *-commutative 92.1%

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

   4. Applied egg-rr 92.1%

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

   5. Taylor expanded in x.re around inf 84.5%

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

      1. mul-1-neg 84.5%

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

      2. unsub-neg 84.5%

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

   7. Applied egg-rr 84.5%

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

      1. pow1 84.5%

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

      2. *-commutative 84.5%

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

      3. associate-*l* 84.4%

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

   9. Applied egg-rr 84.4%

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

       1. unpow1 84.4%

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

       2. associate-*r* 90.9%

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

       3. +-commutative 90.9%

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

   11. Simplified 90.9%

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

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

   1. Initial program 0.0%

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

      1. difference-of-squares 20.7%

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

      2. *-commutative 20.7%

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

   4. Applied egg-rr 20.7%

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

   5. Taylor expanded in x.re around inf 20.7%

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

      1. mul-1-neg 20.7%

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

      2. unsub-neg 20.7%

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

   7. Applied egg-rr 20.7%

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

   8. Taylor expanded in x.re around 0 79.3%

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

      1. mul-1-neg 79.3%

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

   10. Simplified 79.3%

       \[\leadsto \color{blue}{-{x.im}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{-179}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right) - {x.im}^{3}\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(\left(1 - \frac{x.im}{x.re}\right) \cdot \left(x.re + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-{x.im}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + t\_0\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-292} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;-{x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(x.re \cdot x.im\_m\right) \cdot \left(\left(1 - \frac{x.im\_m}{x.re}\right) \cdot \left(x.re + x.im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))
        (t_1 (+ (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m))) t_0)))
   (*
    x.im_s
    (if (or (<= t_1 -1e-292) (not (<= t_1 INFINITY)))
      (- (pow x.im_m 3.0))
      (+
       t_0
       (* (* x.re x.im_m) (* (- 1.0 (/ x.im_m x.re)) (+ x.re x.im_m))))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0;
	double tmp;
	if ((t_1 <= -1e-292) || !(t_1 <= ((double) INFINITY))) {
		tmp = -pow(x_46_im_m, 3.0);
	} else {
		tmp = t_0 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0;
	double tmp;
	if ((t_1 <= -1e-292) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = -Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = t_0 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0
	tmp = 0
	if (t_1 <= -1e-292) or not (t_1 <= math.inf):
		tmp = -math.pow(x_46_im_m, 3.0)
	else:
		tmp = t_0 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + t_0)
	tmp = 0.0
	if ((t_1 <= -1e-292) || !(t_1 <= Inf))
		tmp = Float64(-(x_46_im_m ^ 3.0));
	else
		tmp = Float64(t_0 + Float64(Float64(x_46_re * x_46_im_m) * Float64(Float64(1.0 - Float64(x_46_im_m / x_46_re)) * Float64(x_46_re + x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0;
	tmp = 0.0;
	if ((t_1 <= -1e-292) || ~((t_1 <= Inf)))
		tmp = -(x_46_im_m ^ 3.0);
	else
		tmp = t_0 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$1, -1e-292], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), N[(t$95$0 + N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(N[(1.0 - N[(x$46$im$95$m / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im$95$m), $MachinePrecision]), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.0000000000000001e-292 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 70.9%

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

      1. difference-of-squares 75.6%

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

      2. *-commutative 75.6%

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

   4. Applied egg-rr 75.6%

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

   5. Taylor expanded in x.re around inf 75.6%

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

      1. mul-1-neg 75.6%

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

      2. unsub-neg 75.6%

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

   7. Applied egg-rr 75.6%

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

   8. Taylor expanded in x.re around 0 60.2%

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

      1. mul-1-neg 60.2%

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

   10. Simplified 60.2%

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

   if -1.0000000000000001e-292 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.8%

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

      1. difference-of-squares 94.8%

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

      2. *-commutative 94.8%

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

   4. Applied egg-rr 94.8%

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

   5. Taylor expanded in x.re around inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. unsub-neg 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. pow1 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-*l* 89.8%

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

   9. Applied egg-rr 89.8%

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

       1. unpow1 89.8%

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

       2. associate-*r* 93.3%

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

       3. +-commutative 93.3%

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

   11. Simplified 93.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-292} \lor \neg \left(x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty\right):\\ \;\;\;\;-{x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(\left(1 - \frac{x.im}{x.re}\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right)\\ t_1 := x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 + t\_1 \leq -1 \cdot 10^{-292}:\\ \;\;\;\;t\_0 + x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(x.re \cdot x.im\_m\right) \cdot \left(\left(1 - \frac{x.im\_m}{x.re}\right) \cdot \left(x.re + x.im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m))))
        (t_1 (* x.re (+ (* x.re x.im_m) (* x.re x.im_m)))))
   (*
    x.im_s
    (if (<= (+ t_0 t_1) -1e-292)
      (+ t_0 (* x.re (* (* x.re x.im_m) 2.0)))
      (+
       t_1
       (* (* x.re x.im_m) (* (- 1.0 (/ x.im_m x.re)) (+ x.re x.im_m))))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m));
	double t_1 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m));
	double tmp;
	if ((t_0 + t_1) <= -1e-292) {
		tmp = t_0 + (x_46_re * ((x_46_re * x_46_im_m) * 2.0));
	} else {
		tmp = t_1 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_46im_m * ((x_46re * x_46re) - (x_46im_m * x_46im_m))
    t_1 = x_46re * ((x_46re * x_46im_m) + (x_46re * x_46im_m))
    if ((t_0 + t_1) <= (-1d-292)) then
        tmp = t_0 + (x_46re * ((x_46re * x_46im_m) * 2.0d0))
    else
        tmp = t_1 + ((x_46re * x_46im_m) * ((1.0d0 - (x_46im_m / x_46re)) * (x_46re + x_46im_m)))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m));
	double t_1 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m));
	double tmp;
	if ((t_0 + t_1) <= -1e-292) {
		tmp = t_0 + (x_46_re * ((x_46_re * x_46_im_m) * 2.0));
	} else {
		tmp = t_1 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))
	t_1 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m))
	tmp = 0
	if (t_0 + t_1) <= -1e-292:
		tmp = t_0 + (x_46_re * ((x_46_re * x_46_im_m) * 2.0))
	else:
		tmp = t_1 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)))
	tmp = 0.0
	if (Float64(t_0 + t_1) <= -1e-292)
		tmp = Float64(t_0 + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 2.0)));
	else
		tmp = Float64(t_1 + Float64(Float64(x_46_re * x_46_im_m) * Float64(Float64(1.0 - Float64(x_46_im_m / x_46_re)) * Float64(x_46_re + x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m));
	t_1 = x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m));
	tmp = 0.0;
	if ((t_0 + t_1) <= -1e-292)
		tmp = t_0 + (x_46_re * ((x_46_re * x_46_im_m) * 2.0));
	else
		tmp = t_1 + ((x_46_re * x_46_im_m) * ((1.0 - (x_46_im_m / x_46_re)) * (x_46_re + x_46_im_m)));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[N[(t$95$0 + t$95$1), $MachinePrecision], -1e-292], N[(t$95$0 + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(N[(1.0 - N[(x$46$im$95$m / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im$95$m), $MachinePrecision]), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.0000000000000001e-292

   1. Initial program 92.0%

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

      1. *-commutative 92.0%

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

      2. count-2 92.0%

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

      3. *-commutative 92.0%

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

   4. Applied egg-rr 92.0%

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

   if -1.0000000000000001e-292 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 77.5%

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

      1. difference-of-squares 81.3%

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

      2. *-commutative 81.3%

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in x.re around inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. unsub-neg 77.2%

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

   7. Applied egg-rr 77.2%

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

      1. pow1 77.2%

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

      2. *-commutative 77.2%

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

      3. associate-*l* 77.2%

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

   9. Applied egg-rr 77.2%

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

       1. unpow1 77.2%

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

       2. associate-*r* 80.1%

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

       3. +-commutative 80.1%

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

   11. Simplified 80.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-292}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(\left(1 - \frac{x.im}{x.re}\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 2\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 + x.im\_m \cdot \left(x.re \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re (* (* x.re x.im_m) 2.0))))
   (*
    x.im_s
    (if (<= x.re 1.35e+154)
      (+ (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m))) t_0)
      (+ t_0 (* x.im_m (* x.re x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_re * ((x_46_re * x_46_im_m) * 2.0);
	double tmp;
	if (x_46_re <= 1.35e+154) {
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0;
	} else {
		tmp = t_0 + (x_46_im_m * (x_46_re * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re * ((x_46re * x_46im_m) * 2.0d0)
    if (x_46re <= 1.35d+154) then
        tmp = (x_46im_m * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) + t_0
    else
        tmp = t_0 + (x_46im_m * (x_46re * x_46im_m))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_re * ((x_46_re * x_46_im_m) * 2.0);
	double tmp;
	if (x_46_re <= 1.35e+154) {
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0;
	} else {
		tmp = t_0 + (x_46_im_m * (x_46_re * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = x_46_re * ((x_46_re * x_46_im_m) * 2.0)
	tmp = 0
	if x_46_re <= 1.35e+154:
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0
	else:
		tmp = t_0 + (x_46_im_m * (x_46_re * x_46_im_m))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 2.0))
	tmp = 0.0
	if (x_46_re <= 1.35e+154)
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + t_0);
	else
		tmp = Float64(t_0 + Float64(x_46_im_m * Float64(x_46_re * x_46_im_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = x_46_re * ((x_46_re * x_46_im_m) * 2.0);
	tmp = 0.0;
	if (x_46_re <= 1.35e+154)
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + t_0;
	else
		tmp = t_0 + (x_46_im_m * (x_46_re * x_46_im_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$re, 1.35e+154], N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 + N[(x$46$im$95$m * N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.35000000000000003e154

   1. Initial program 86.4%

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

      1. *-commutative 86.4%

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

      2. count-2 86.4%

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

      3. *-commutative 86.4%

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

   4. Applied egg-rr 86.4%

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

   if 1.35000000000000003e154 < x.re

   1. Initial program 46.7%

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

      1. difference-of-squares 55.8%

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

      2. *-commutative 55.8%

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

   4. Applied egg-rr 55.8%

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

   5. Taylor expanded in x.re around 0 45.4%

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

      1. *-commutative 46.7%

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

      2. count-2 46.7%

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

      3. *-commutative 46.7%

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

   7. Applied egg-rr 45.4%

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

   8. Taylor expanded in x.re around inf 45.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right) + x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 380:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 2\right) + x.im\_m \cdot \left(x.im\_m \cdot \left(x.re - x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 380.0)
    (+ (* x.re (* (* x.re x.im_m) 2.0)) (* x.im_m (* x.im_m (- x.re x.im_m))))
    (* 3.0 (* (* x.re x.re) x.im_m)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 380.0) {
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)));
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 380.0d0) then
        tmp = (x_46re * ((x_46re * x_46im_m) * 2.0d0)) + (x_46im_m * (x_46im_m * (x_46re - x_46im_m)))
    else
        tmp = 3.0d0 * ((x_46re * x_46re) * x_46im_m)
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 380.0) {
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)));
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 380.0:
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)))
	else:
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im_m)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 380.0)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 2.0)) + Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re - x_46_im_m))));
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 380.0)
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)));
	else
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im_m);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 380.0], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 380

   1. Initial program 88.0%

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

      1. difference-of-squares 90.0%

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

      2. *-commutative 90.0%

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

   4. Applied egg-rr 90.0%

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

   5. Taylor expanded in x.re around 0 79.4%

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

      1. *-commutative 88.0%

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

      2. count-2 88.0%

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

      3. *-commutative 88.0%

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

   7. Applied egg-rr 79.4%

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

   if 380 < x.re

   1. Initial program 66.4%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in x.re around inf 64.7%

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

      1. pow2 64.7%

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

   7. Applied egg-rr 64.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 380:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right) + x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * ((x_46re * ((x_46re * x_46im_m) + (x_46re * x_46im_m))) + (x_46im_m * ((x_46re - x_46im_m) * (x_46re + x_46im_m))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

   1. difference-of-squares 85.4%

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

   2. *-commutative 85.4%

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

4. Applied egg-rr 85.4%

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

5. Final simplification 85.4%

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

Alternative 7: 50.4% accurate, 2.7× speedup

Math

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

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (* x.im_s (* 3.0 (* (* x.re x.re) x.im_m))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * (3.0 * ((x_46_re * x_46_re) * x_46_im_m));
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (3.0d0 * ((x_46re * x_46re) * x_46im_m))
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * (3.0 * ((x_46_re * x_46_re) * x_46_im_m));
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	return x_46_im_s * (3.0 * ((x_46_re * x_46_re) * x_46_im_m))

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * Float64(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im_m)))
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * (3.0 * ((x_46_re * x_46_re) * x_46_im_m));
end

Wolfram

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

Derivation

1. Initial program 83.0%

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

3. Simplified 84.7%

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

5. Taylor expanded in x.re around inf 48.6%

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

   1. pow2 48.6%

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

7. Applied egg-rr 48.6%

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

8. Final simplification 48.6%

   \[\leadsto 3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right) \]
9. Add Preprocessing

Alternative 8: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (-3.0d0)
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	return x_46_im_s * -3.0

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * -3.0)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * -3.0;
end

Wolfram

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

Derivation

1. Initial program 83.0%

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

3. Simplified 84.7%

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

5. Taylor expanded in x.re around 0 61.7%

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]

7. Simplified 2.7%

   \[\leadsto \color{blue}{-3} \]
8. Add Preprocessing

Developer Target 1: 91.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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