math.cube on complex, imaginary part

Percentage Accurate: 82.2% → 99.8%

Time: 9.2s

Alternatives: 7

Speedup: 1.2×

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

Specification

Math

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

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: 82.2% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+84}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im\_m \cdot 3\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -3\\ \end{array} \end{array} \]

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.im_m 5e+84)
    (- (* x.re (* x.re (* x.im_m 3.0))) (pow x.im_m 3.0))
    (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -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 tmp;
	if (x_46_im_m <= 5e+84) {
		tmp = (x_46_re * (x_46_re * (x_46_im_m * 3.0))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	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_46im_m <= 5d+84) then
        tmp = (x_46re * (x_46re * (x_46im_m * 3.0d0))) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-3.0d0)
    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_im_m <= 5e+84) {
		tmp = (x_46_re * (x_46_re * (x_46_im_m * 3.0))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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):
	tmp = 0
	if x_46_im_m <= 5e+84:
		tmp = (x_46_re * (x_46_re * (x_46_im_m * 3.0))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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)
	tmp = 0.0
	if (x_46_im_m <= 5e+84)
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im_m * 3.0))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -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)
	tmp = 0.0;
	if (x_46_im_m <= 5e+84)
		tmp = (x_46_re * (x_46_re * (x_46_im_m * 3.0))) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+84], N[(N[(x$46$re * N[(x$46$re * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000001e84

   1. Initial program 88.6%

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

   3. Simplified 92.0%

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

   if 5.0000000000000001e84 < x.im

   1. Initial program 78.0%

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

   3. Taylor expanded in x.re around 0 78.0%

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

   5. Simplified 88.0%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+84}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right)\\ t_1 := t\_0 + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{+216}:\\ \;\;\;\;t\_0 + x.re \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot -3 + x.re \cdot \left(x.im\_m + x.im\_m \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -3\\ \end{array} \end{array} \end{array} \]

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.im_m (- (* x.re x.re) (* x.im_m x.im_m))))
        (t_1 (+ t_0 (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (*
    x.im_s
    (if (<= t_1 1e+216)
      (+ t_0 (* x.re (* (* x.im_m x.re) 2.0)))
      (if (<= t_1 INFINITY)
        (* x.re (+ (* x.im_m -3.0) (* x.re (+ x.im_m (* x.im_m 2.0)))))
        (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -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_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m));
	double t_1 = t_0 + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= 1e+216) {
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_re * ((x_46_im_m * -3.0) + (x_46_re * (x_46_im_m + (x_46_im_m * 2.0))));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m));
	double t_1 = t_0 + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= 1e+216) {
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re * ((x_46_im_m * -3.0) + (x_46_re * (x_46_im_m + (x_46_im_m * 2.0))));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))
	t_1 = t_0 + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))
	tmp = 0
	if t_1 <= 1e+216:
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0))
	elif t_1 <= math.inf:
		tmp = x_46_re * ((x_46_im_m * -3.0) + (x_46_re * (x_46_im_m + (x_46_im_m * 2.0))))
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(t_0 + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_1 <= 1e+216)
		tmp = Float64(t_0 + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_re * Float64(Float64(x_46_im_m * -3.0) + Float64(x_46_re * Float64(x_46_im_m + Float64(x_46_im_m * 2.0)))));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -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_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m));
	t_1 = t_0 + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	tmp = 0.0;
	if (t_1 <= 1e+216)
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	elseif (t_1 <= Inf)
		tmp = x_46_re * ((x_46_im_m * -3.0) + (x_46_re * (x_46_im_m + (x_46_im_m * 2.0))));
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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$46$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[(t$95$0 + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, 1e+216], N[(t$95$0 + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x$46$re * N[(N[(x$46$im$95$m * -3.0), $MachinePrecision] + N[(x$46$re * N[(x$46$im$95$m + N[(x$46$im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -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)) < 1e216

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. count-2 96.8%

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

      3. *-commutative 96.8%

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

   4. Applied egg-rr 96.8%

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

   if 1e216 < (+.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 88.1%

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

      1. *-commutative 88.1%

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

      2. count-2 88.1%

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

      3. *-commutative 88.1%

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

   4. Applied egg-rr 88.1%

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

      1. difference-of-squares 88.1%

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

   6. Applied egg-rr 88.1%

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

   8. Simplified 47.8%

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

   9. Taylor expanded in x.im around 0 46.5%

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

   10. Taylor expanded in x.re around 0 58.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x.re around 0 0.0%

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

   5. Simplified 39.1%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 10^{+216}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.im \cdot -3 + x.re \cdot \left(x.im + x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1400000:\\ \;\;\;\;x.re \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot 2\right) + x.im\_m \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -3\\ \end{array} \end{array} \]

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.im_m 1400000.0)
    (+ (* x.re (* (* x.im_m x.re) 2.0)) (* x.im_m (* x.re x.re)))
    (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -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 tmp;
	if (x_46_im_m <= 1400000.0) {
		tmp = (x_46_re * ((x_46_im_m * x_46_re) * 2.0)) + (x_46_im_m * (x_46_re * x_46_re));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	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_46im_m <= 1400000.0d0) then
        tmp = (x_46re * ((x_46im_m * x_46re) * 2.0d0)) + (x_46im_m * (x_46re * x_46re))
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-3.0d0)
    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_im_m <= 1400000.0) {
		tmp = (x_46_re * ((x_46_im_m * x_46_re) * 2.0)) + (x_46_im_m * (x_46_re * x_46_re));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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):
	tmp = 0
	if x_46_im_m <= 1400000.0:
		tmp = (x_46_re * ((x_46_im_m * x_46_re) * 2.0)) + (x_46_im_m * (x_46_re * x_46_re))
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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)
	tmp = 0.0
	if (x_46_im_m <= 1400000.0)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) * 2.0)) + Float64(x_46_im_m * Float64(x_46_re * x_46_re)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -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)
	tmp = 0.0;
	if (x_46_im_m <= 1400000.0)
		tmp = (x_46_re * ((x_46_im_m * x_46_re) * 2.0)) + (x_46_im_m * (x_46_re * x_46_re));
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1400000.0], N[(N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.4e6

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. count-2 87.9%

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

      3. *-commutative 87.9%

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

   4. Applied egg-rr 87.9%

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

      1. difference-of-squares 91.0%

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

   6. Applied egg-rr 91.0%

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

   8. Simplified 56.1%

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

   9. Taylor expanded in x.im around 0 57.2%

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

   10. Taylor expanded in x.re around inf 66.3%

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

   if 1.4e6 < x.im

   1. Initial program 82.2%

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

   3. Taylor expanded in x.re around 0 82.2%

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

   5. Simplified 88.8%

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

      1. difference-of-squares 98.5%

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

      2. *-commutative 98.5%

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

   7. Applied egg-rr 98.5%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1400000:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) + x.im \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1400000:\\ \;\;\;\;\left(x.im\_m \cdot 3\right) \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -3\\ \end{array} \end{array} \]

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.im_m 1400000.0)
    (* (* x.im_m 3.0) (* x.re x.re))
    (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -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 tmp;
	if (x_46_im_m <= 1400000.0) {
		tmp = (x_46_im_m * 3.0) * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	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_46im_m <= 1400000.0d0) then
        tmp = (x_46im_m * 3.0d0) * (x_46re * x_46re)
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-3.0d0)
    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_im_m <= 1400000.0) {
		tmp = (x_46_im_m * 3.0) * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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):
	tmp = 0
	if x_46_im_m <= 1400000.0:
		tmp = (x_46_im_m * 3.0) * (x_46_re * x_46_re)
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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)
	tmp = 0.0
	if (x_46_im_m <= 1400000.0)
		tmp = Float64(Float64(x_46_im_m * 3.0) * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -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)
	tmp = 0.0;
	if (x_46_im_m <= 1400000.0)
		tmp = (x_46_im_m * 3.0) * (x_46_re * x_46_re);
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -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_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1400000.0], N[(N[(x$46$im$95$m * 3.0), $MachinePrecision] * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.4e6

   1. Initial program 87.9%

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

      1. +-commutative 87.9%

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

      2. *-commutative 87.9%

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

      3. sqr-neg 87.9%

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

      4. fma-define 88.4%

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

      5. remove-double-neg 88.4%

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

      6. *-commutative 88.4%

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

      7. distribute-neg-out 88.4%

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

      8. distribute-lft-neg-out 88.4%

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

      9. distribute-lft-neg-out 88.4%

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

      10. *-commutative 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in x.re around inf 66.3%

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

      1. pow2 66.3%

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

   7. Applied egg-rr 66.3%

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

   8. Taylor expanded in x.im around 0 66.3%

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

      1. *-commutative 66.3%

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

   10. Simplified 66.3%

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

   if 1.4e6 < x.im

   1. Initial program 82.2%

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

   3. Taylor expanded in x.re around 0 82.2%

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

   5. Simplified 88.8%

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

      1. difference-of-squares 98.5%

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

      2. *-commutative 98.5%

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

   7. Applied egg-rr 98.5%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1400000:\\ \;\;\;\;\left(x.im \cdot 3\right) \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.25 \cdot 10^{+214}:\\ \;\;\;\;\left(x.im\_m \cdot 3\right) \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im\_m \cdot -3\right)\\ \end{array} \end{array} \]

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.im_m 1.25e+214)
    (* (* x.im_m 3.0) (* x.re x.re))
    (* (* x.re x.re) (* 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 tmp;
	if (x_46_im_m <= 1.25e+214) {
		tmp = (x_46_im_m * 3.0) * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_im_m * -3.0);
	}
	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_46im_m <= 1.25d+214) then
        tmp = (x_46im_m * 3.0d0) * (x_46re * x_46re)
    else
        tmp = (x_46re * x_46re) * (x_46im_m * (-3.0d0))
    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_im_m <= 1.25e+214) {
		tmp = (x_46_im_m * 3.0) * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_re * x_46_re) * (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):
	tmp = 0
	if x_46_im_m <= 1.25e+214:
		tmp = (x_46_im_m * 3.0) * (x_46_re * x_46_re)
	else:
		tmp = (x_46_re * x_46_re) * (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)
	tmp = 0.0
	if (x_46_im_m <= 1.25e+214)
		tmp = Float64(Float64(x_46_im_m * 3.0) * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * 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)
	tmp = 0.0;
	if (x_46_im_m <= 1.25e+214)
		tmp = (x_46_im_m * 3.0) * (x_46_re * x_46_re);
	else
		tmp = (x_46_re * x_46_re) * (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_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.25e+214], N[(N[(x$46$im$95$m * 3.0), $MachinePrecision] * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.24999999999999988e214

   1. Initial program 88.7%

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

      1. +-commutative 88.7%

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

      2. *-commutative 88.7%

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

      3. sqr-neg 88.7%

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

      4. fma-define 89.2%

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

      5. remove-double-neg 89.2%

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

      6. *-commutative 89.2%

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

      7. distribute-neg-out 89.2%

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

      8. distribute-lft-neg-out 89.2%

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

      9. distribute-lft-neg-out 89.2%

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

      10. *-commutative 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in x.re around inf 60.7%

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

      1. pow2 60.7%

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

   7. Applied egg-rr 60.7%

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

   8. Taylor expanded in x.im around 0 60.7%

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

      1. *-commutative 60.7%

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

   10. Simplified 60.7%

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

   if 1.24999999999999988e214 < x.im

   1. Initial program 61.9%

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

      1. +-commutative 61.9%

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

      2. *-commutative 61.9%

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

      3. sqr-neg 61.9%

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

      4. fma-define 66.7%

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

      5. remove-double-neg 66.7%

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

      6. *-commutative 66.7%

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

      7. distribute-neg-out 66.7%

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

      8. distribute-lft-neg-out 66.7%

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

      9. distribute-lft-neg-out 66.7%

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

      10. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in x.re around inf 5.3%

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

      1. pow2 5.3%

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

   7. Applied egg-rr 5.3%

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

      1. add-sqr-sqrt 5.3%

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

      2. pow2 5.3%

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

      3. *-un-lft-identity 5.3%

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

      4. distribute-rgt-out 5.3%

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

      5. metadata-eval 5.3%

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

   9. Applied egg-rr 5.3%

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

       1. unpow2 5.3%

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

       2. sqrt-unprod 4.8%

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

       3. swap-sqr 4.8%

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

       4. metadata-eval 4.8%

          \[\leadsto \left(x.re \cdot x.re\right) \cdot \sqrt{\left(x.im \cdot x.im\right) \cdot \color{blue}{9}} \]

       5. metadata-eval 4.8%

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

       6. swap-sqr 4.8%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 35.2%

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

   11. Applied egg-rr 35.2%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.25 \cdot 10^{+214}:\\ \;\;\;\;\left(x.im \cdot 3\right) \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 23.3% accurate, 2.7× speedup

Math

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

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 -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 * ((x_46_re * x_46_re) * (x_46_im_m * -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 * ((x_46re * x_46re) * (x_46im_m * (-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 * ((x_46_re * x_46_re) * (x_46_im_m * -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 * ((x_46_re * x_46_re) * (x_46_im_m * -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 * Float64(Float64(x_46_re * x_46_re) * Float64(x_46_im_m * -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 * ((x_46_re * x_46_re) * (x_46_im_m * -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 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.5%

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

   1. +-commutative 86.5%

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

   2. *-commutative 86.5%

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

   3. sqr-neg 86.5%

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

   4. fma-define 87.3%

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

   5. remove-double-neg 87.3%

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

   6. *-commutative 87.3%

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

   7. distribute-neg-out 87.3%

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

   8. distribute-lft-neg-out 87.3%

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

   9. distribute-lft-neg-out 87.3%

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

   10. *-commutative 87.3%

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

3. Simplified 87.3%

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

5. Taylor expanded in x.re around inf 56.1%

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

   1. pow2 56.1%

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

7. Applied egg-rr 56.1%

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

   1. add-sqr-sqrt 28.3%

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

   2. pow2 28.3%

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

   3. *-un-lft-identity 28.3%

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

   4. distribute-rgt-out 28.3%

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

   5. metadata-eval 28.3%

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

9. Applied egg-rr 28.3%

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

    1. unpow2 28.3%

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

    2. sqrt-unprod 37.0%

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

    3. swap-sqr 37.0%

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

    4. metadata-eval 37.0%

       \[\leadsto \left(x.re \cdot x.re\right) \cdot \sqrt{\left(x.im \cdot x.im\right) \cdot \color{blue}{9}} \]

    5. metadata-eval 37.0%

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

    6. swap-sqr 37.0%

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

    7. sqrt-unprod 10.8%

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

    8. add-sqr-sqrt 21.5%

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

11. Applied egg-rr 21.5%

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

Alternative 7: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot 10 \end{array} \]

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 10.0))

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 * 10.0;
}

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 * 10.0d0
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 * 10.0;
}

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 * 10.0

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 * 10.0)
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 * 10.0;
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 * 10.0), $MachinePrecision]

Derivation

1. Initial program 86.5%

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

3. Simplified 87.3%

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

   1. sub-neg 87.3%

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

   2. associate-*r* 87.4%

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

   3. associate-*l* 87.3%

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

   4. flip-+ 24.4%

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

   5. swap-sqr 24.4%

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

   6. associate-*r* 23.3%

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

   7. associate-*r* 23.2%

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

   8. swap-sqr 14.6%

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

   9. pow2 14.6%

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

   10. pow2 14.6%

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

   11. pow-prod-up 14.6%

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

   12. metadata-eval 14.6%

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

   13. pow2 14.6%

       \[\leadsto \frac{\left({x.re}^{4} \cdot \color{blue}{{x.im}^{2}}\right) \cdot \left(3 \cdot 3\right) - \left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right)}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 - \left(-{x.im}^{3}\right)} \]

   14. metadata-eval 14.6%

       \[\leadsto \frac{\left({x.re}^{4} \cdot {x.im}^{2}\right) \cdot \color{blue}{9} - \left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right)}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 - \left(-{x.im}^{3}\right)} \]

6. Applied egg-rr 14.6%

   \[\leadsto \color{blue}{\frac{\left({x.re}^{4} \cdot {x.im}^{2}\right) \cdot 9 - \left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right)}{{x.re}^{2} \cdot \left(x.im \cdot 3\right) - \left(-{x.im}^{3}\right)}} \]

8. Simplified 2.7%

   \[\leadsto \color{blue}{10} \]
9. Add Preprocessing

Developer Target 1: 91.6% 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 2024185 
(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)))