math.cube on complex, imaginary part

Percentage Accurate: 82.9% → 99.6%

Time: 8.9s

Alternatives: 13

Speedup: 1.4×

• 44.6% 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: 82.9% 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: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2 \cdot 10^{+76}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot 3\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 2e+76)
    (- (* x.re_m (* x.im_m (* x.re_m 3.0))) (pow x.im_m 3.0))
    (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2e+76) {
		tmp = (x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 2d+76) then
        tmp = (x_46re_m * (x_46im_m * (x_46re_m * 3.0d0))) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2e+76) {
		tmp = (x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 2e+76:
		tmp = (x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2e+76)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_re_m * 3.0))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 2e+76)
		tmp = (x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0))) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 2e+76], N[(N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$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[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.0000000000000001e76

   1. Initial program 82.1%

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

   3. Simplified 89.9%

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

   if 2.0000000000000001e76 < x.im

   1. Initial program 56.8%

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

      1. difference-of-squares 68.2%

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

      2. *-commutative 68.2%

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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in x.re around 0 61.4%

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

   6. Taylor expanded in x.re around 0 61.4%

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

   8. Simplified 93.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2 \cdot 10^{+76}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right)\\ t_1 := t\_0 + x.re\_m \cdot \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+228}:\\ \;\;\;\;t\_0 + x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m \cdot x.re\_m, x.re\_m \cdot 3, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m))))
        (t_1 (+ t_0 (* x.re_m (+ (* x.im_m x.re_m) (* x.im_m x.re_m))))))
   (*
    x.im_s
    (if (<= t_1 5e+228)
      (+ t_0 (* x.re_m (* (* x.im_m x.re_m) 2.0)))
      (if (<= t_1 INFINITY)
        (fma (* x.im_m x.re_m) (* x.re_m 3.0) -1.0)
        (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0))))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	double t_1 = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	double tmp;
	if (t_1 <= 5e+228) {
		tmp = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x_46_im_m * x_46_re_m), (x_46_re_m * 3.0), -1.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m))))
	tmp = 0.0
	if (t_1 <= 5e+228)
		tmp = Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = fma(Float64(x_46_im_m * x_46_re_m), Float64(x_46_re_m * 3.0), -1.0);
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $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$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, 5e+228], N[(t$95$0 + N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * N[(x$46$re$95$m * 3.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $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)) < 5e228

   1. Initial program 92.5%

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

      1. *-commutative 92.5%

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

      2. count-2 92.5%

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

      3. *-commutative 92.5%

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

   4. Applied egg-rr 92.5%

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

   if 5e228 < (+.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 85.8%

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

   3. Simplified 98.0%

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

      1. associate-*r* 98.0%

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

      2. fma-neg 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in x.im around 0 98.0%

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

   9. Simplified 54.0%

      \[\leadsto \mathsf{fma}\left(x.re \cdot x.im, x.re \cdot 3, \color{blue}{-1}\right) \]

   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. Step-by-step derivation

      1. difference-of-squares 27.0%

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

      2. *-commutative 27.0%

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

   4. Applied egg-rr 27.0%

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

   5. Taylor expanded in x.re around 0 16.2%

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

   6. Taylor expanded in x.re around 0 16.2%

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

   8. Simplified 89.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 5 \cdot 10^{+228}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot x.re, x.re \cdot 3, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right)\\ t_1 := t\_0 + x.re\_m \cdot \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+228}:\\ \;\;\;\;t\_0 + x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m \cdot \left(x.re\_m \cdot 3\right), x.re\_m, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m))))
        (t_1 (+ t_0 (* x.re_m (+ (* x.im_m x.re_m) (* x.im_m x.re_m))))))
   (*
    x.im_s
    (if (<= t_1 5e+228)
      (+ t_0 (* x.re_m (* (* x.im_m x.re_m) 2.0)))
      (if (<= t_1 INFINITY)
        (fma (* x.im_m (* x.re_m 3.0)) x.re_m -1.0)
        (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0))))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	double t_1 = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	double tmp;
	if (t_1 <= 5e+228) {
		tmp = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x_46_im_m * (x_46_re_m * 3.0)), x_46_re_m, -1.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m))))
	tmp = 0.0
	if (t_1 <= 5e+228)
		tmp = Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = fma(Float64(x_46_im_m * Float64(x_46_re_m * 3.0)), x_46_re_m, -1.0);
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $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$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, 5e+228], N[(t$95$0 + N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x$46$im$95$m * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m + -1.0), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $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)) < 5e228

   1. Initial program 92.5%

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

      1. *-commutative 92.5%

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

      2. count-2 92.5%

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

      3. *-commutative 92.5%

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

   4. Applied egg-rr 92.5%

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

   if 5e228 < (+.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 85.8%

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

   3. Simplified 98.0%

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

      1. *-commutative 98.0%

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

      2. fma-neg 98.0%

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

      3. associate-*r* 98.0%

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

      4. *-commutative 98.0%

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

      5. associate-*l* 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x.im around 0 97.9%

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

   9. Simplified 53.9%

      \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im \cdot 3\right), x.re, \color{blue}{-1}\right) \]
   10. Step-by-step derivation

       1. *-un-lft-identity 53.9%

          \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(x.re \cdot \left(x.im \cdot 3\right), x.re, -1\right)} \]

       2. associate-*r* 53.9%

          \[\leadsto 1 \cdot \mathsf{fma}\left(\color{blue}{\left(x.re \cdot x.im\right) \cdot 3}, x.re, -1\right) \]

       3. *-commutative 53.9%

          \[\leadsto 1 \cdot \mathsf{fma}\left(\color{blue}{\left(x.im \cdot x.re\right)} \cdot 3, x.re, -1\right) \]

       4. associate-*l* 54.0%

          \[\leadsto 1 \cdot \mathsf{fma}\left(\color{blue}{x.im \cdot \left(x.re \cdot 3\right)}, x.re, -1\right) \]

   11. Applied egg-rr 54.0%

       \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 3\right), x.re, -1\right)} \]

   13. Simplified 54.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 3\right), x.re, -1\right)} \]

   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. Step-by-step derivation

      1. difference-of-squares 27.0%

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

      2. *-commutative 27.0%

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

   4. Applied egg-rr 27.0%

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

   5. Taylor expanded in x.re around 0 16.2%

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

   6. Taylor expanded in x.re around 0 16.2%

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

   8. Simplified 89.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 5 \cdot 10^{+228}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 3\right), x.re, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2 \cdot 10^{+76}:\\ \;\;\;\;x.re\_m \cdot \left(3 \cdot \left(x.im\_m \cdot x.re\_m\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 2e+76)
    (- (* x.re_m (* 3.0 (* x.im_m x.re_m))) (pow x.im_m 3.0))
    (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2e+76) {
		tmp = (x_46_re_m * (3.0 * (x_46_im_m * x_46_re_m))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 2d+76) then
        tmp = (x_46re_m * (3.0d0 * (x_46im_m * x_46re_m))) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2e+76) {
		tmp = (x_46_re_m * (3.0 * (x_46_im_m * x_46_re_m))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 2e+76:
		tmp = (x_46_re_m * (3.0 * (x_46_im_m * x_46_re_m))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2e+76)
		tmp = Float64(Float64(x_46_re_m * Float64(3.0 * Float64(x_46_im_m * x_46_re_m))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 2e+76)
		tmp = (x_46_re_m * (3.0 * (x_46_im_m * x_46_re_m))) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 2e+76], N[(N[(x$46$re$95$m * N[(3.0 * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.0000000000000001e76

   1. Initial program 82.1%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in x.im around 0 89.9%

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

   if 2.0000000000000001e76 < x.im

   1. Initial program 56.8%

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

      1. difference-of-squares 68.2%

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

      2. *-commutative 68.2%

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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in x.re around 0 61.4%

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

   6. Taylor expanded in x.re around 0 61.4%

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

   8. Simplified 93.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2 \cdot 10^{+76}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 0.5× speedup

Math

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

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))))
   (*
    x.im_s
    (if (<=
         (+ t_0 (* x.re_m (+ (* x.im_m x.re_m) (* x.im_m x.re_m))))
         INFINITY)
      (+ t_0 (* x.re_m (* (* x.im_m x.re_m) 2.0)))
      (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0)))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	double tmp;
	if ((t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))) <= ((double) INFINITY)) {
		tmp = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	double tmp;
	if ((t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))
	tmp = 0
	if (t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))) <= math.inf:
		tmp = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0))
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)))
	tmp = 0.0
	if (Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)))) <= Inf)
		tmp = Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	tmp = 0.0;
	if ((t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))) <= Inf)
		tmp = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[N[(t$95$0 + N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 + N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.9%

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

      1. *-commutative 90.9%

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

      2. count-2 90.9%

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

      3. *-commutative 90.9%

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

   4. Applied egg-rr 90.9%

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

   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 27.0%

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

      2. *-commutative 27.0%

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

   4. Applied egg-rr 27.0%

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

   5. Taylor expanded in x.re around 0 16.2%

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

   6. Taylor expanded in x.re around 0 16.2%

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

   8. Simplified 89.2%

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

4. Final simplification 90.7%

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

Alternative 6: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.4 \cdot 10^{-96}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot \left(x.im\_m \cdot 3\right)\\ \mathbf{elif}\;x.im\_m \leq 2 \cdot 10^{+76}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) - x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.4e-96)
    (* (* x.re_m x.re_m) (* x.im_m 3.0))
    (if (<= x.im_m 2e+76)
      (-
       (* x.re_m (+ (* x.im_m x.re_m) (* x.im_m x.re_m)))
       (* x.im_m (* x.im_m x.im_m)))
      (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0)))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.4e-96) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else if (x_46_im_m <= 2e+76) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m))) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 1.4d-96) then
        tmp = (x_46re_m * x_46re_m) * (x_46im_m * 3.0d0)
    else if (x_46im_m <= 2d+76) then
        tmp = (x_46re_m * ((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m))) - (x_46im_m * (x_46im_m * x_46im_m))
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.4e-96) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else if (x_46_im_m <= 2e+76) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m))) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 1.4e-96:
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0)
	elif x_46_im_m <= 2e+76:
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m))) - (x_46_im_m * (x_46_im_m * x_46_im_m))
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.4e-96)
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * Float64(x_46_im_m * 3.0));
	elseif (x_46_im_m <= 2e+76)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m))) - Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 1.4e-96)
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	elseif (x_46_im_m <= 2e+76)
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m))) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.4e-96], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 2e+76], N[(N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 1.40000000000000008e-96

   1. Initial program 79.4%

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

   3. Simplified 88.1%

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

      1. associate-*r* 88.1%

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

      2. fma-neg 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in x.re around inf 59.4%

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

      1. associate-*r* 59.4%

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

      2. *-commutative 59.4%

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

      3. *-commutative 59.4%

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

   9. Simplified 59.4%

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

       1. unpow2 59.4%

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

   11. Applied egg-rr 59.4%

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

   if 1.40000000000000008e-96 < x.im < 2.0000000000000001e76

   1. Initial program 97.1%

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

      1. difference-of-squares 97.1%

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

      2. *-commutative 97.1%

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

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in x.re around 0 85.4%

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

   6. Taylor expanded in x.re around 0 92.6%

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

   8. Simplified 92.6%

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

   if 2.0000000000000001e76 < x.im

   1. Initial program 56.8%

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

      1. difference-of-squares 68.2%

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

      2. *-commutative 68.2%

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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in x.re around 0 61.4%

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

   6. Taylor expanded in x.re around 0 61.4%

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

   8. Simplified 93.2%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.4 \cdot 10^{-96}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{+76}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.8 \cdot 10^{-98}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot \left(x.im\_m \cdot 3\right)\\ \mathbf{elif}\;x.im\_m \leq 7.2 \cdot 10^{+75}:\\ \;\;\;\;t\_0 + x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -3\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (* x.im_m (- x.re_m x.im_m)))))
   (*
    x.im_s
    (if (<= x.im_m 3.8e-98)
      (* (* x.re_m x.re_m) (* x.im_m 3.0))
      (if (<= x.im_m 7.2e+75)
        (+ t_0 (* x.re_m (* (* x.im_m x.re_m) 2.0)))
        (+ t_0 -3.0))))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double t_0 = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	double tmp;
	if (x_46_im_m <= 3.8e-98) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else if (x_46_im_m <= 7.2e+75) {
		tmp = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = t_0 + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))
    if (x_46im_m <= 3.8d-98) then
        tmp = (x_46re_m * x_46re_m) * (x_46im_m * 3.0d0)
    else if (x_46im_m <= 7.2d+75) then
        tmp = t_0 + (x_46re_m * ((x_46im_m * x_46re_m) * 2.0d0))
    else
        tmp = t_0 + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	double t_0 = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	double tmp;
	if (x_46_im_m <= 3.8e-98) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else if (x_46_im_m <= 7.2e+75) {
		tmp = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = t_0 + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	t_0 = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))
	tmp = 0
	if x_46_im_m <= 3.8e-98:
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0)
	elif x_46_im_m <= 7.2e+75:
		tmp = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0))
	else:
		tmp = t_0 + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)))
	tmp = 0.0
	if (x_46_im_m <= 3.8e-98)
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * Float64(x_46_im_m * 3.0));
	elseif (x_46_im_m <= 7.2e+75)
		tmp = Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)));
	else
		tmp = Float64(t_0 + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	t_0 = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	tmp = 0.0;
	if (x_46_im_m <= 3.8e-98)
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	elseif (x_46_im_m <= 7.2e+75)
		tmp = t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	else
		tmp = t_0 + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 3.8e-98], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 7.2e+75], N[(t$95$0 + N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -3.0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.im < 3.8000000000000003e-98

   1. Initial program 79.4%

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

   3. Simplified 88.1%

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

      1. associate-*r* 88.1%

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

      2. fma-neg 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in x.re around inf 59.4%

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

      1. associate-*r* 59.4%

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

      2. *-commutative 59.4%

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

      3. *-commutative 59.4%

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

   9. Simplified 59.4%

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

       1. unpow2 59.4%

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

   11. Applied egg-rr 59.4%

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

   if 3.8000000000000003e-98 < x.im < 7.2e75

   1. Initial program 97.1%

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

      1. difference-of-squares 97.1%

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

      2. *-commutative 97.1%

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

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in x.re around 0 85.4%

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

      1. *-commutative 97.1%

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

      2. count-2 97.1%

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

      3. *-commutative 97.1%

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

   7. Applied egg-rr 85.4%

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

   if 7.2e75 < x.im

   1. Initial program 56.8%

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

      1. difference-of-squares 68.2%

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

      2. *-commutative 68.2%

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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in x.re around 0 61.4%

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

   6. Taylor expanded in x.re around 0 61.4%

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

   8. Simplified 93.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.8 \cdot 10^{-98}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{elif}\;x.im \leq 7.2 \cdot 10^{+75}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 200:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot \left(x.im\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 200.0)
    (* (* x.re_m x.re_m) (* x.im_m 3.0))
    (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 200.0) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 200.0d0) then
        tmp = (x_46re_m * x_46re_m) * (x_46im_m * 3.0d0)
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 200.0) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 200.0:
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0)
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 200.0)
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * Float64(x_46_im_m * 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 200.0)
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 200.0], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 200

   1. Initial program 81.1%

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

   3. Simplified 89.3%

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

      1. associate-*r* 89.3%

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

      2. fma-neg 92.8%

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

   6. Applied egg-rr 92.8%

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

   7. Taylor expanded in x.re around inf 57.0%

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

      1. associate-*r* 57.0%

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

      2. *-commutative 57.0%

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

      3. *-commutative 57.0%

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

   9. Simplified 57.0%

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

       1. unpow2 57.0%

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

   11. Applied egg-rr 57.0%

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

   if 200 < x.im

   1. Initial program 66.0%

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

      1. difference-of-squares 74.9%

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

      2. *-commutative 74.9%

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

   4. Applied egg-rr 74.9%

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

   5. Taylor expanded in x.re around 0 64.6%

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

   6. Taylor expanded in x.re around 0 64.6%

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

   8. Simplified 85.6%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 200:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 6.8 \cdot 10^{+117}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot \left(x.im\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im\_m \cdot \left(x.im\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 6.8e+117)
    (* (* x.re_m x.re_m) (* x.im_m 3.0))
    (- -1.0 (* x.im_m (* x.im_m x.re_m))))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 6.8e+117) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_im_m * x_46_re_m));
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 6.8d+117) then
        tmp = (x_46re_m * x_46re_m) * (x_46im_m * 3.0d0)
    else
        tmp = (-1.0d0) - (x_46im_m * (x_46im_m * x_46re_m))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 6.8e+117) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_im_m * x_46_re_m));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 6.8e+117:
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0)
	else:
		tmp = -1.0 - (x_46_im_m * (x_46_im_m * x_46_re_m))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 6.8e+117)
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * Float64(x_46_im_m * 3.0));
	else
		tmp = Float64(-1.0 - Float64(x_46_im_m * Float64(x_46_im_m * x_46_re_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 6.8e+117)
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	else
		tmp = -1.0 - (x_46_im_m * (x_46_im_m * x_46_re_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 6.8e+117], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 6.8000000000000002e117

   1. Initial program 82.9%

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

   3. Simplified 89.9%

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

      1. associate-*r* 89.9%

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

      2. fma-neg 93.0%

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

   6. Applied egg-rr 93.0%

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

   7. Taylor expanded in x.re around inf 56.5%

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

      1. associate-*r* 56.5%

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

      2. *-commutative 56.5%

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

      3. *-commutative 56.5%

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

   9. Simplified 56.5%

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

       1. unpow2 56.5%

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

   11. Applied egg-rr 56.5%

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

   if 6.8000000000000002e117 < x.im

   1. Initial program 45.7%

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

      1. difference-of-squares 60.0%

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

      2. *-commutative 60.0%

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

   4. Applied egg-rr 60.0%

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

   5. Taylor expanded in x.re around 0 85.7%

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

   7. Simplified 41.2%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 6.8 \cdot 10^{+117}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * ((x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0));
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * ((x_46re_m * x_46re_m) * (x_46im_m * 3.0d0))
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * ((x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0));
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	return x_46_im_s * ((x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0))

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(x_46_re_m * x_46_re_m) * Float64(x_46_im_m * 3.0)))
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = x_46_im_s * ((x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0));
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

3. Simplified 83.8%

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

   1. associate-*r* 83.8%

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

   2. fma-neg 88.1%

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

6. Applied egg-rr 88.1%

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

7. Taylor expanded in x.re around inf 50.8%

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

   1. associate-*r* 50.8%

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

   2. *-commutative 50.8%

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

   3. *-commutative 50.8%

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

9. Simplified 50.8%

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

    1. unpow2 50.8%

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

11. Applied egg-rr 50.8%

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

Alternative 11: 50.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)));
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (3.0d0 * (x_46im_m * (x_46re_m * x_46re_m)))
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)));
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)))

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_re_m))))
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)));
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

3. Simplified 83.8%

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

5. Taylor expanded in x.re around inf 50.8%

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

   1. unpow2 50.8%

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

7. Applied egg-rr 50.8%

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

Alternative 12: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot 0.1 \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m) :precision binary64 (* x.im_s 0.1))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * 0.1;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * 0.1d0
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * 0.1;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	return x_46_im_s * 0.1

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	return Float64(x_46_im_s * 0.1)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = x_46_im_s * 0.1;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * 0.1), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

3. Simplified 83.8%

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

   1. sub-neg 83.8%

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

   2. flip3-+ 16.7%

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

   3. associate-*r* 16.6%

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

   4. *-commutative 16.6%

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

   5. associate-*l* 16.6%

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

   6. associate-*r* 15.9%

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

   7. associate-*l* 15.9%

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

   8. pow2 15.9%

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

6. Applied egg-rr 8.6%

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

8. Simplified 2.8%

   \[\leadsto \color{blue}{0.1} \]
9. Add Preprocessing

Alternative 13: 2.7% accurate, 19.0× speedup

Math

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

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m) :precision binary64 (* x.im_s -1.0))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * -1.0;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (-1.0d0)
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * -1.0;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	return x_46_im_s * -1.0

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	return Float64(x_46_im_s * -1.0)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = x_46_im_s * -1.0;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * -1.0), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

   1. difference-of-squares 81.7%

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

   2. *-commutative 81.7%

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

4. Applied egg-rr 81.7%

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

5. Taylor expanded in x.re around 0 57.5%

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

7. Simplified 16.7%

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

8. Taylor expanded in x.im around 0 2.8%

   \[\leadsto \color{blue}{-1} \]
9. 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 2024145 
(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)))