math.cube on complex, imaginary part

Percentage Accurate: 83.3% → 99.6%

Time: 9.1s

Alternatives: 11

Speedup: 1.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.3% 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

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot 3\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

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 1.5e+82)
    (- (* x.re_m (* x.re_m (* x.im_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 <= 1.5e+82) {
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_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 <= 1.5d+82) then
        tmp = (x_46re_m * (x_46re_m * (x_46im_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 <= 1.5e+82) {
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_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 <= 1.5e+82:
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_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 <= 1.5e+82)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im_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 <= 1.5e+82)
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_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, 1.5e+82], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(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 < 1.49999999999999995e82

   1. Initial program 89.1%

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

   3. Simplified 94.4%

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

   if 1.49999999999999995e82 < x.im

   1. Initial program 78.6%

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

      1. difference-of-squares 85.7%

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

      2. *-commutative 85.7%

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

   4. Applied egg-rr 85.7%

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

   5. Taylor expanded in x.re around 0 75.0%

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

   6. Taylor expanded in x.re around 0 75.0%

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

   8. Simplified 89.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 6 \cdot 10^{-139}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot 3\right)\right)\\ \mathbf{elif}\;x.im\_m \leq 2.05 \cdot 10^{+78}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right) + x.im\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.im\_m + x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

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 6e-139)
    (* x.re_m (* x.im_m (* x.re_m 3.0)))
    (if (<= x.im_m 2.05e+78)
      (+
       (* x.re_m (* (* x.im_m x.re_m) 2.0))
       (* x.im_m (* (- x.re_m x.im_m) (+ x.im_m x.re_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 <= 6e-139) {
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0));
	} else if (x_46_im_m <= 2.05e+78) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) + (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_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 <= 6d-139) then
        tmp = x_46re_m * (x_46im_m * (x_46re_m * 3.0d0))
    else if (x_46im_m <= 2.05d+78) then
        tmp = (x_46re_m * ((x_46im_m * x_46re_m) * 2.0d0)) + (x_46im_m * ((x_46re_m - x_46im_m) * (x_46im_m + x_46re_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 <= 6e-139) {
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0));
	} else if (x_46_im_m <= 2.05e+78) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) + (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_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 <= 6e-139:
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0))
	elif x_46_im_m <= 2.05e+78:
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) + (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_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 <= 6e-139)
		tmp = Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_re_m * 3.0)));
	elseif (x_46_im_m <= 2.05e+78)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)) + Float64(x_46_im_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_im_m + x_46_re_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 <= 6e-139)
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0));
	elseif (x_46_im_m <= 2.05e+78)
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) + (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_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, 6e-139], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 2.05e+78], N[(N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $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 < 5.9999999999999998e-139

   1. Initial program 85.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in x.re around inf 54.3%

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

      1. *-commutative 54.3%

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

      2. *-commutative 54.3%

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

      3. associate-*l* 54.3%

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

      4. pow2 54.3%

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

      5. add-sqr-sqrt 19.0%

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

      6. swap-sqr 22.8%

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

      7. unpow2 22.8%

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

   7. Applied egg-rr 22.8%

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

      1. unpow2 22.8%

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

      2. *-commutative 22.8%

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

      3. *-commutative 22.8%

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

      4. swap-sqr 19.0%

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

      5. add-sqr-sqrt 54.3%

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

      6. associate-*r* 62.0%

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

   9. Applied egg-rr 62.0%

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

   10. Taylor expanded in x.im around 0 62.0%

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

       1. *-commutative 62.0%

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

       2. associate-*l* 62.0%

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

   12. Simplified 62.0%

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

   if 5.9999999999999998e-139 < x.im < 2.0499999999999998e78

   1. Initial program 99.7%

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

      1. difference-of-squares 99.7%

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

      2. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. count-2 99.7%

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

      3. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   if 2.0499999999999998e78 < x.im

   1. Initial program 78.9%

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

      1. difference-of-squares 86.0%

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

      2. *-commutative 86.0%

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

   4. Applied egg-rr 86.0%

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

   5. Taylor expanded in x.re around 0 75.4%

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

   6. Taylor expanded in x.re around 0 75.4%

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

   8. Simplified 89.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 6 \cdot 10^{-139}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 2.05 \cdot 10^{+78}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 25000000:\\ \;\;\;\;\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot \left(1 + \frac{x.im\_m}{x.re\_m}\right)\right)\right) + x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;-3 + x.im\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.im\_m + x.re\_m\right)\right)\\ \end{array} \end{array} \]

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 25000000.0)
    (+
     (* (- x.re_m x.im_m) (* x.re_m (* x.im_m (+ 1.0 (/ x.im_m x.re_m)))))
     (* x.re_m (* (* x.im_m x.re_m) 2.0)))
    (+ -3.0 (* x.im_m (* (- x.re_m 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 <= 25000000.0) {
		tmp = ((x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_im_m * (1.0 + (x_46_im_m / x_46_re_m))))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - 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 <= 25000000.0d0) then
        tmp = ((x_46re_m - x_46im_m) * (x_46re_m * (x_46im_m * (1.0d0 + (x_46im_m / x_46re_m))))) + (x_46re_m * ((x_46im_m * x_46re_m) * 2.0d0))
    else
        tmp = (-3.0d0) + (x_46im_m * ((x_46re_m - 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 <= 25000000.0) {
		tmp = ((x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_im_m * (1.0 + (x_46_im_m / x_46_re_m))))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - 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 <= 25000000.0:
		tmp = ((x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_im_m * (1.0 + (x_46_im_m / x_46_re_m))))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0))
	else:
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - 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 <= 25000000.0)
		tmp = Float64(Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m * Float64(x_46_im_m * Float64(1.0 + Float64(x_46_im_m / x_46_re_m))))) + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)));
	else
		tmp = Float64(-3.0 + Float64(x_46_im_m * Float64(Float64(x_46_re_m - 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 <= 25000000.0)
		tmp = ((x_46_re_m - x_46_im_m) * (x_46_re_m * (x_46_im_m * (1.0 + (x_46_im_m / x_46_re_m))))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	else
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - 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, 25000000.0], N[(N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$im$95$m * N[(1.0 + N[(x$46$im$95$m / x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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[(-3.0 + N[(x$46$im$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.5e7

   1. Initial program 88.5%

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

      1. difference-of-squares 90.6%

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

      2. *-commutative 90.6%

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

   4. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. count-2 90.6%

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

      3. *-commutative 90.6%

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

   6. Applied egg-rr 90.6%

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

   7. Taylor expanded in x.re around inf 87.6%

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

      1. pow1 87.6%

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

      2. associate-*l* 93.6%

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

      3. associate-*l* 92.6%

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

   9. Applied egg-rr 92.6%

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

       1. unpow1 92.6%

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

       2. *-commutative 92.6%

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

   11. Simplified 92.6%

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

   if 2.5e7 < x.im

   1. Initial program 82.1%

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

      1. difference-of-squares 88.1%

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

      2. *-commutative 88.1%

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

   4. Applied egg-rr 88.1%

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

   5. Taylor expanded in x.re around 0 88.1%

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

   7. Simplified 100.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 25000000:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(1 + \frac{x.im}{x.re}\right)\right)\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;-3 + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.5 \cdot 10^{-86}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot 3\right)\right)\\ \mathbf{elif}\;x.im\_m \leq 2.05 \cdot 10^{+78}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right) - x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

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 1.5e-86)
    (* x.re_m (* x.im_m (* x.re_m 3.0)))
    (if (<= x.im_m 2.05e+78)
      (- (* x.re_m (* (* x.im_m x.re_m) 2.0)) (* 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 <= 1.5e-86) {
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0));
	} else if (x_46_im_m <= 2.05e+78) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (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 <= 1.5d-86) then
        tmp = x_46re_m * (x_46im_m * (x_46re_m * 3.0d0))
    else if (x_46im_m <= 2.05d+78) then
        tmp = (x_46re_m * ((x_46im_m * x_46re_m) * 2.0d0)) - (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 <= 1.5e-86) {
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0));
	} else if (x_46_im_m <= 2.05e+78) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (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 <= 1.5e-86:
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0))
	elif x_46_im_m <= 2.05e+78:
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (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 <= 1.5e-86)
		tmp = Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_re_m * 3.0)));
	elseif (x_46_im_m <= 2.05e+78)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)) - 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 <= 1.5e-86)
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0));
	elseif (x_46_im_m <= 2.05e+78)
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (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, 1.5e-86], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 2.05e+78], N[(N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $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 < 1.5e-86

   1. Initial program 86.6%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x.re around inf 58.3%

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

      1. *-commutative 58.3%

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

      2. *-commutative 58.3%

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

      3. associate-*l* 58.3%

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

      4. pow2 58.3%

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

      5. add-sqr-sqrt 26.0%

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

      6. swap-sqr 29.5%

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

      7. unpow2 29.5%

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

   7. Applied egg-rr 29.5%

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

      1. unpow2 29.5%

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

      2. *-commutative 29.5%

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

      3. *-commutative 29.5%

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

      4. swap-sqr 26.0%

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

      5. add-sqr-sqrt 58.3%

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

      6. associate-*r* 65.2%

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

   9. Applied egg-rr 65.2%

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

   10. Taylor expanded in x.im around 0 65.2%

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

       1. *-commutative 65.2%

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

       2. associate-*l* 65.2%

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

   12. Simplified 65.2%

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

   if 1.5e-86 < x.im < 2.0499999999999998e78

   1. Initial program 99.7%

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

      1. difference-of-squares 99.7%

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

      2. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. count-2 99.7%

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

      3. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x.re around 0 77.0%

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

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

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

   if 2.0499999999999998e78 < x.im

   1. Initial program 78.9%

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

      1. difference-of-squares 86.0%

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

      2. *-commutative 86.0%

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

   4. Applied egg-rr 86.0%

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

   5. Taylor expanded in x.re around 0 75.4%

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

   6. Taylor expanded in x.re around 0 75.4%

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

   8. Simplified 89.5%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{-86}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 2.05 \cdot 10^{+78}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 + x.im\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.im\_m + x.re\_m\right)\right)\\ \end{array} \end{array} \]

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 3.5e-5)
    (* x.re_m (* x.re_m (* x.im_m 3.0)))
    (+ -3.0 (* x.im_m (* (- x.re_m 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 <= 3.5e-5) {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0));
	} else {
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - 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 <= 3.5d-5) then
        tmp = x_46re_m * (x_46re_m * (x_46im_m * 3.0d0))
    else
        tmp = (-3.0d0) + (x_46im_m * ((x_46re_m - 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 <= 3.5e-5) {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0));
	} else {
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - 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 <= 3.5e-5:
		tmp = x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0))
	else:
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - 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 <= 3.5e-5)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im_m * 3.0)));
	else
		tmp = Float64(-3.0 + Float64(x_46_im_m * Float64(Float64(x_46_re_m - 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 <= 3.5e-5)
		tmp = x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0));
	else
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - 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, 3.5e-5], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 + N[(x$46$im$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.4999999999999997e-5

   1. Initial program 88.4%

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

   3. Simplified 94.0%

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

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

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

      1. *-commutative 56.1%

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

      2. *-commutative 56.1%

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

      3. associate-*l* 56.2%

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

      4. pow2 56.2%

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

      5. add-sqr-sqrt 28.2%

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

      6. swap-sqr 31.2%

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

      7. unpow2 31.2%

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

   7. Applied egg-rr 31.2%

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

      1. unpow2 31.2%

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

      2. *-commutative 31.2%

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

      3. *-commutative 31.2%

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

      4. swap-sqr 28.2%

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

      5. add-sqr-sqrt 56.2%

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

      6. associate-*r* 62.2%

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

   9. Applied egg-rr 62.2%

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

   if 3.4999999999999997e-5 < x.im

   1. Initial program 82.6%

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

      1. difference-of-squares 88.4%

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

      2. *-commutative 88.4%

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

   4. Applied egg-rr 88.4%

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

   5. Taylor expanded in x.re around 0 88.4%

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

   7. Simplified 97.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 75000000:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

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 75000000.0)
    (* 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 <= 75000000.0) {
		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 <= 75000000.0d0) 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 <= 75000000.0) {
		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 <= 75000000.0:
		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 <= 75000000.0)
		tmp = Float64(x_46_re_m * Float64(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 <= 75000000.0)
		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, 75000000.0], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * 3.0), $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 x.im < 7.5e7

   1. Initial program 88.5%

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

   3. Simplified 94.0%

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

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

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

      1. *-commutative 56.1%

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

      2. *-commutative 56.1%

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

      3. associate-*l* 56.1%

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

      4. pow2 56.1%

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

      5. add-sqr-sqrt 28.4%

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

      6. swap-sqr 31.4%

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

      7. unpow2 31.4%

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

   7. Applied egg-rr 31.4%

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

      1. unpow2 31.4%

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

      2. *-commutative 31.4%

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

      3. *-commutative 31.4%

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

      4. swap-sqr 28.4%

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

      5. add-sqr-sqrt 56.1%

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

      6. associate-*r* 62.1%

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

   9. Applied egg-rr 62.1%

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

   if 7.5e7 < x.im

   1. Initial program 82.1%

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

      1. difference-of-squares 88.1%

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

      2. *-commutative 88.1%

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

   4. Applied egg-rr 88.1%

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

   5. Taylor expanded in x.re around 0 79.1%

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

   6. Taylor expanded in x.re around 0 79.1%

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

   8. Simplified 84.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 75000000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.4 \cdot 10^{+146}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 + x.im\_m \cdot \left(x.im\_m \cdot -3\right)\\ \end{array} \end{array} \]

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 3.4e+146)
    (* x.re_m (* x.re_m (* x.im_m 3.0)))
    (+ -3.0 (* x.im_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 <= 3.4e+146) {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0));
	} else {
		tmp = -3.0 + (x_46_im_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 <= 3.4d+146) then
        tmp = x_46re_m * (x_46re_m * (x_46im_m * 3.0d0))
    else
        tmp = (-3.0d0) + (x_46im_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 <= 3.4e+146) {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0));
	} else {
		tmp = -3.0 + (x_46_im_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 <= 3.4e+146:
		tmp = x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0))
	else:
		tmp = -3.0 + (x_46_im_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 <= 3.4e+146)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im_m * 3.0)));
	else
		tmp = Float64(-3.0 + Float64(x_46_im_m * Float64(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 <= 3.4e+146)
		tmp = x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0));
	else
		tmp = -3.0 + (x_46_im_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, 3.4e+146], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 + N[(x$46$im$95$m * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.39999999999999991e146

   1. Initial program 89.5%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in x.re around inf 55.8%

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

      1. *-commutative 55.8%

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

      2. *-commutative 55.8%

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

      3. associate-*l* 55.9%

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

      4. pow2 55.9%

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

      5. add-sqr-sqrt 31.7%

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

      6. swap-sqr 34.3%

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

      7. unpow2 34.3%

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

   7. Applied egg-rr 34.3%

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

      1. unpow2 34.3%

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

      2. *-commutative 34.3%

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

      3. *-commutative 34.3%

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

      4. swap-sqr 31.7%

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

      5. add-sqr-sqrt 55.9%

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

      6. associate-*r* 61.1%

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

   9. Applied egg-rr 61.1%

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

   if 3.39999999999999991e146 < x.im

   1. Initial program 71.8%

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

      1. difference-of-squares 82.1%

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

      2. *-commutative 82.1%

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

   4. Applied egg-rr 82.1%

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

   5. Taylor expanded in x.re around 0 74.4%

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

   6. Taylor expanded in x.re around 0 74.4%

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

   8. Simplified 92.3%

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

   9. Taylor expanded in x.re around 0 89.7%

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

   11. Simplified 85.5%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.4 \cdot 10^{+146}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 + x.im \cdot \left(x.im \cdot -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.9% accurate, 2.7× speedup

Math

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

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 (* 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(x_46_re_m * Float64(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[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.8%

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

3. Simplified 89.0%

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

5. Taylor expanded in x.re around inf 49.0%

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

   1. *-commutative 49.0%

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

   2. *-commutative 49.0%

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

   3. associate-*l* 49.0%

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

   4. pow2 49.0%

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

   5. add-sqr-sqrt 28.5%

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

   6. swap-sqr 30.7%

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

   7. unpow2 30.7%

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

7. Applied egg-rr 30.7%

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

   1. unpow2 30.7%

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

   2. *-commutative 30.7%

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

   3. *-commutative 30.7%

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

   4. swap-sqr 28.5%

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

   5. add-sqr-sqrt 49.0%

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

   6. associate-*r* 53.4%

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

9. Applied egg-rr 53.4%

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

10. Final simplification 53.4%

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

Alternative 9: 55.9% accurate, 2.7× speedup

Math

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

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 (* x.re_m (* x.im_m (* x.re_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_im_m * (x_46_re_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_46im_m * (x_46re_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_im_m * (x_46_re_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_im_m * (x_46_re_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(x_46_re_m * Float64(x_46_im_m * Float64(x_46_re_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_im_m * (x_46_re_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[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.8%

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

3. Simplified 89.0%

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

5. Taylor expanded in x.re around inf 49.0%

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

   1. *-commutative 49.0%

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

   2. *-commutative 49.0%

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

   3. associate-*l* 49.0%

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

   4. pow2 49.0%

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

   5. add-sqr-sqrt 28.5%

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

   6. swap-sqr 30.7%

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

   7. unpow2 30.7%

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

7. Applied egg-rr 30.7%

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

   1. unpow2 30.7%

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

   2. *-commutative 30.7%

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

   3. *-commutative 30.7%

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

   4. swap-sqr 28.5%

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

   5. add-sqr-sqrt 49.0%

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

   6. associate-*r* 53.4%

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

9. Applied egg-rr 53.4%

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

10. Taylor expanded in x.im around 0 53.4%

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

    1. *-commutative 53.4%

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

    2. associate-*l* 53.4%

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

12. Simplified 53.4%

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

13. Final simplification 53.4%

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

Alternative 10: 50.3% accurate, 2.7× speedup

Math

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

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 86.8%

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

3. Simplified 89.0%

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

5. Taylor expanded in x.re around inf 49.0%

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

   1. pow2 49.0%

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

7. Applied egg-rr 49.0%

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

Alternative 11: 2.7% accurate, 19.0× speedup

Math

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

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))

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;
}

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)
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;
}

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

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 * -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 * -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 * -3.0), $MachinePrecision]

Derivation

1. Initial program 86.8%

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

   1. difference-of-squares 89.9%

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

   2. *-commutative 89.9%

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

4. Applied egg-rr 89.9%

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

5. Taylor expanded in x.re around 0 69.7%

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

6. Taylor expanded in x.re around 0 69.0%

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

8. Simplified 45.1%

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

9. Taylor expanded in x.im around 0 2.6%

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

Developer Target 1: 91.8% 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 2024165 
(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)))