math.cube on complex, imaginary part

Percentage Accurate: 82.8% → 99.6%

Time: 7.8s

Alternatives: 11

Speedup: 0.9×

• 44.0% 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: 82.8% 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

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+70}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re \cdot 3\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5e+70)
    (- (* x.re (* x.im_m (* x.re 3.0))) (pow x.im_m 3.0))
    (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -1.0))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+70) {
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 5d+70) then
        tmp = (x_46re * (x_46im_m * (x_46re * 3.0d0))) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-1.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+70) {
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 5e+70:
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+70)
		tmp = Float64(Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re * 3.0))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -1.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 5e+70)
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+70], N[(N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000002e70

   1. Initial program 86.0%

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

   3. Simplified 92.6%

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

   if 5.0000000000000002e70 < x.im

   1. Initial program 77.6%

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

      1. difference-of-squares 83.7%

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

      2. *-commutative 83.7%

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

   4. Applied egg-rr 83.7%

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

      1. *-commutative 83.7%

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

      2. *-un-lft-identity 83.7%

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

      3. *-un-lft-identity 83.7%

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

      4. distribute-rgt-out 83.7%

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

      5. metadata-eval 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. add-log-exp 40.8%

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

      2. *-commutative 40.8%

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

      3. add-log-exp 83.7%

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

      4. count-2 83.7%

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

      5. *-commutative 83.7%

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

      6. expm1-log1p-u 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

      7. expm1-undefine 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

      8. *-commutative 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

      9. *-commutative 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

      10. flip-+ 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

      11. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

      12. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

   10. Simplified 100.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+70}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4.3 \cdot 10^{-87}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) + \left(x.re - x.im\_m\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \mathbf{elif}\;x.im\_m \leq 5 \cdot 10^{+69}:\\ \;\;\;\;t\_0 + x.re \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re)))))
   (*
    x.im_s
    (if (<= x.im_m 4.3e-87)
      (+
       (* x.re (+ (* x.im_m x.re) (* x.im_m x.re)))
       (* (- x.re x.im_m) (* x.im_m x.re)))
      (if (<= x.im_m 5e+69)
        (+ t_0 (* x.re (* (* x.im_m x.re) 2.0)))
        (+ t_0 -1.0))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	double tmp;
	if (x_46_im_m <= 4.3e-87) {
		tmp = (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re))) + ((x_46_re - x_46_im_m) * (x_46_im_m * x_46_re));
	} else if (x_46_im_m <= 5e+69) {
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = t_0 + -1.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))
    if (x_46im_m <= 4.3d-87) then
        tmp = (x_46re * ((x_46im_m * x_46re) + (x_46im_m * x_46re))) + ((x_46re - x_46im_m) * (x_46im_m * x_46re))
    else if (x_46im_m <= 5d+69) then
        tmp = t_0 + (x_46re * ((x_46im_m * x_46re) * 2.0d0))
    else
        tmp = t_0 + (-1.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	double tmp;
	if (x_46_im_m <= 4.3e-87) {
		tmp = (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re))) + ((x_46_re - x_46_im_m) * (x_46_im_m * x_46_re));
	} else if (x_46_im_m <= 5e+69) {
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = t_0 + -1.0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))
	tmp = 0
	if x_46_im_m <= 4.3e-87:
		tmp = (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re))) + ((x_46_re - x_46_im_m) * (x_46_im_m * x_46_re))
	elif x_46_im_m <= 5e+69:
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0))
	else:
		tmp = t_0 + -1.0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re)))
	tmp = 0.0
	if (x_46_im_m <= 4.3e-87)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))) + Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m * x_46_re)));
	elseif (x_46_im_m <= 5e+69)
		tmp = Float64(t_0 + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) * 2.0)));
	else
		tmp = Float64(t_0 + -1.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	tmp = 0.0;
	if (x_46_im_m <= 4.3e-87)
		tmp = (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re))) + ((x_46_re - x_46_im_m) * (x_46_im_m * x_46_re));
	elseif (x_46_im_m <= 5e+69)
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	else
		tmp = t_0 + -1.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4.3e-87], N[(N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 5e+69], N[(t$95$0 + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.im < 4.29999999999999995e-87

   1. Initial program 85.3%

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

      1. difference-of-squares 85.3%

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

      2. *-commutative 85.3%

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

   4. Applied egg-rr 85.3%

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

      1. *-commutative 85.3%

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

      2. distribute-rgt-in 84.2%

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

      3. distribute-lft-in 80.4%

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

   6. Applied egg-rr 80.4%

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

   8. Simplified 84.2%

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

      1. +-commutative 84.2%

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

      2. distribute-rgt-in 80.4%

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

      3. *-commutative 80.4%

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

      4. associate-*l* 80.4%

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

      5. pow2 80.4%

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

      6. *-commutative 80.4%

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

      7. associate-*l* 87.9%

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

      8. *-commutative 87.9%

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

   10. Applied egg-rr 87.9%

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

   12. Simplified 92.7%

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

   13. Taylor expanded in x.im around 0 72.0%

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

   if 4.29999999999999995e-87 < x.im < 5.00000000000000036e69

   1. Initial program 91.8%

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

      1. difference-of-squares 91.8%

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

      2. *-commutative 91.8%

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

   4. Applied egg-rr 91.8%

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

      1. *-commutative 91.8%

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

      2. *-un-lft-identity 91.8%

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

      3. *-un-lft-identity 91.8%

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

      4. distribute-rgt-out 91.8%

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

      5. metadata-eval 91.8%

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

   6. Applied egg-rr 91.8%

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

   if 5.00000000000000036e69 < x.im

   1. Initial program 77.6%

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

      1. difference-of-squares 83.7%

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

      2. *-commutative 83.7%

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

   4. Applied egg-rr 83.7%

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

      1. *-commutative 83.7%

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

      2. *-un-lft-identity 83.7%

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

      3. *-un-lft-identity 83.7%

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

      4. distribute-rgt-out 83.7%

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

      5. metadata-eval 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. add-log-exp 40.8%

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

      2. *-commutative 40.8%

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

      3. add-log-exp 83.7%

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

      4. count-2 83.7%

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

      5. *-commutative 83.7%

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

      6. expm1-log1p-u 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

      7. expm1-undefine 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

      8. *-commutative 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

      9. *-commutative 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

      10. flip-+ 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

      11. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

      12. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

   10. Simplified 100.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.3 \cdot 10^{-87}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) + \left(x.re - x.im\right) \cdot \left(x.im \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+69}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 10^{+71}:\\ \;\;\;\;\left(x.re - x.im\_m\right) \cdot \left(x.im\_m \cdot \left(x.im\_m + x.re\right)\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1e+71)
    (+
     (* (- x.re x.im_m) (* x.im_m (+ x.im_m x.re)))
     (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))
    (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -1.0))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1e+71) {
		tmp = ((x_46_re - x_46_im_m) * (x_46_im_m * (x_46_im_m + x_46_re))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 1d+71) then
        tmp = ((x_46re - x_46im_m) * (x_46im_m * (x_46im_m + x_46re))) + (x_46re * ((x_46im_m * x_46re) + (x_46im_m * x_46re)))
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-1.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1e+71) {
		tmp = ((x_46_re - x_46_im_m) * (x_46_im_m * (x_46_im_m + x_46_re))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 1e+71:
		tmp = ((x_46_re - x_46_im_m) * (x_46_im_m * (x_46_im_m + x_46_re))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1e+71)
		tmp = Float64(Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m * Float64(x_46_im_m + x_46_re))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -1.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 1e+71)
		tmp = ((x_46_re - x_46_im_m) * (x_46_im_m * (x_46_im_m + x_46_re))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1e+71], N[(N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1e71

   1. Initial program 86.0%

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

      1. difference-of-squares 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. Step-by-step derivation

      1. *-commutative 86.0%

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

      2. distribute-rgt-in 85.0%

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

      3. distribute-lft-in 81.6%

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

   6. Applied egg-rr 81.6%

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

   8. Simplified 85.0%

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

      1. +-commutative 85.0%

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

      2. distribute-rgt-in 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-*l* 81.6%

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

      5. pow2 81.6%

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

      6. *-commutative 81.6%

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

      7. associate-*l* 89.1%

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

      8. *-commutative 89.1%

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

   10. Applied egg-rr 89.1%

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

   12. Simplified 93.5%

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

   if 1e71 < x.im

   1. Initial program 77.6%

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

      1. difference-of-squares 83.7%

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

      2. *-commutative 83.7%

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

   4. Applied egg-rr 83.7%

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

      1. *-commutative 83.7%

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

      2. *-un-lft-identity 83.7%

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

      3. *-un-lft-identity 83.7%

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

      4. distribute-rgt-out 83.7%

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

      5. metadata-eval 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. add-log-exp 40.8%

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

      2. *-commutative 40.8%

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

      3. add-log-exp 83.7%

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

      4. count-2 83.7%

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

      5. *-commutative 83.7%

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

      6. expm1-log1p-u 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

      7. expm1-undefine 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

      8. *-commutative 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

      9. *-commutative 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

      10. flip-+ 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

      11. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

      12. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

   10. Simplified 100.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 10^{+71}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.im + x.re\right)\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.55 \cdot 10^{+71}:\\ \;\;\;\;t\_0 + x.re \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re)))))
   (*
    x.im_s
    (if (<= x.im_m 1.55e+71)
      (+ t_0 (* x.re (* (* x.im_m x.re) 2.0)))
      (+ t_0 -1.0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	double tmp;
	if (x_46_im_m <= 1.55e+71) {
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = t_0 + -1.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))
    if (x_46im_m <= 1.55d+71) then
        tmp = t_0 + (x_46re * ((x_46im_m * x_46re) * 2.0d0))
    else
        tmp = t_0 + (-1.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	double tmp;
	if (x_46_im_m <= 1.55e+71) {
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = t_0 + -1.0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))
	tmp = 0
	if x_46_im_m <= 1.55e+71:
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0))
	else:
		tmp = t_0 + -1.0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re)))
	tmp = 0.0
	if (x_46_im_m <= 1.55e+71)
		tmp = Float64(t_0 + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) * 2.0)));
	else
		tmp = Float64(t_0 + -1.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	tmp = 0.0;
	if (x_46_im_m <= 1.55e+71)
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	else
		tmp = t_0 + -1.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.55e+71], N[(t$95$0 + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.55000000000000009e71

   1. Initial program 86.0%

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

      1. difference-of-squares 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. Step-by-step derivation

      1. *-commutative 86.0%

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

      2. *-un-lft-identity 86.0%

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

      3. *-un-lft-identity 86.0%

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

      4. distribute-rgt-out 86.0%

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

      5. metadata-eval 86.0%

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

   6. Applied egg-rr 86.0%

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

   if 1.55000000000000009e71 < x.im

   1. Initial program 77.6%

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

      1. difference-of-squares 83.7%

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

      2. *-commutative 83.7%

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

   4. Applied egg-rr 83.7%

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

      1. *-commutative 83.7%

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

      2. *-un-lft-identity 83.7%

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

      3. *-un-lft-identity 83.7%

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

      4. distribute-rgt-out 83.7%

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

      5. metadata-eval 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. add-log-exp 40.8%

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

      2. *-commutative 40.8%

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

      3. add-log-exp 83.7%

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

      4. count-2 83.7%

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

      5. *-commutative 83.7%

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

      6. expm1-log1p-u 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

      7. expm1-undefine 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

      8. *-commutative 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

      9. *-commutative 83.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

      10. flip-+ 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

      11. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

      12. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

   10. Simplified 100.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.55 \cdot 10^{+71}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 0.0038:\\ \;\;\;\;t\_0 - x.im\_m \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re)))))
   (* x.im_s (if (<= x.im_m 0.0038) (- t_0 (* x.im_m x.re)) (+ t_0 -1.0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	double tmp;
	if (x_46_im_m <= 0.0038) {
		tmp = t_0 - (x_46_im_m * x_46_re);
	} else {
		tmp = t_0 + -1.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))
    if (x_46im_m <= 0.0038d0) then
        tmp = t_0 - (x_46im_m * x_46re)
    else
        tmp = t_0 + (-1.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	double tmp;
	if (x_46_im_m <= 0.0038) {
		tmp = t_0 - (x_46_im_m * x_46_re);
	} else {
		tmp = t_0 + -1.0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))
	tmp = 0
	if x_46_im_m <= 0.0038:
		tmp = t_0 - (x_46_im_m * x_46_re)
	else:
		tmp = t_0 + -1.0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re)))
	tmp = 0.0
	if (x_46_im_m <= 0.0038)
		tmp = Float64(t_0 - Float64(x_46_im_m * x_46_re));
	else
		tmp = Float64(t_0 + -1.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	tmp = 0.0;
	if (x_46_im_m <= 0.0038)
		tmp = t_0 - (x_46_im_m * x_46_re);
	else
		tmp = t_0 + -1.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 0.0038], N[(t$95$0 - N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 0.00379999999999999999

   1. Initial program 85.3%

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

      1. difference-of-squares 85.3%

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

      2. *-commutative 85.3%

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

   4. Applied egg-rr 85.3%

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

      1. *-commutative 85.3%

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

      2. *-un-lft-identity 85.3%

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

      3. *-un-lft-identity 85.3%

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

      4. distribute-rgt-out 85.3%

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

      5. metadata-eval 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. *-commutative 85.3%

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

      2. count-2 85.3%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \cdot x.re \]

      5. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{0}{\color{blue}{0}} \cdot x.re \]

      6. metadata-eval 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{0 \cdot 0}}{0} \cdot x.re \]

      7. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)} \cdot 0}{0} \cdot x.re \]

      8. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right) \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}}{0} \cdot x.re \]

      9. *-commutative 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}{0} \cdot x.re \]

      10. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}{0} \cdot x.re \]

      11. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)}{0} \cdot x.re \]

      12. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(x.im \cdot x.re\right)\right)}{0} \cdot x.re \]

      13. metadata-eval 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right)}{\color{blue}{0 \cdot 0}} \cdot x.re \]

      14. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right)}{\color{blue}{\left(x.re \cdot x.im - x.re \cdot x.im\right)} \cdot 0} \cdot x.re \]

      15. +-inverses 0.0%

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

      16. *-commutative 0.0%

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

      17. *-commutative 0.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(\frac{0}{0} \cdot \left(x.im \cdot x.re\right) + \frac{0}{0} \cdot \left(x.im \cdot x.re\right)\right)} \cdot x.re \]

   10. Simplified 60.8%

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

   if 0.00379999999999999999 < x.im

   1. Initial program 81.3%

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

      1. difference-of-squares 86.4%

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

      2. *-commutative 86.4%

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

   4. Applied egg-rr 86.4%

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

      1. *-commutative 86.4%

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

      2. *-un-lft-identity 86.4%

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

      3. *-un-lft-identity 86.4%

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

      4. distribute-rgt-out 86.4%

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

      5. metadata-eval 86.4%

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

   6. Applied egg-rr 86.4%

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

      1. add-log-exp 47.7%

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

      2. *-commutative 47.7%

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

      3. add-log-exp 86.4%

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

      4. count-2 86.4%

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

      5. *-commutative 86.4%

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

      6. expm1-log1p-u 86.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

      7. expm1-undefine 86.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

      8. *-commutative 86.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

      9. *-commutative 86.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

      10. flip-+ 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

      11. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

      12. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

   10. Simplified 97.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 0.0038:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) - x.im \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.4 \cdot 10^{-77}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) - x.re\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 3.4e-77)
    (* x.re (* x.im_m (+ x.re -1.0)))
    (- (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) x.re))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3.4e-77) {
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - x_46_re;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 3.4d-77) then
        tmp = x_46re * (x_46im_m * (x_46re + (-1.0d0)))
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) - x_46re
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3.4e-77) {
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - x_46_re;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 3.4e-77:
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0))
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - x_46_re
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 3.4e-77)
		tmp = Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re + -1.0)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) - x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 3.4e-77)
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0));
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - x_46_re;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 3.4e-77], N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.39999999999999983e-77

   1. Initial program 85.5%

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

      1. difference-of-squares 85.5%

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

      2. *-commutative 85.5%

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

   4. Applied egg-rr 85.5%

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

      1. *-commutative 85.5%

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

      2. *-un-lft-identity 85.5%

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

      3. *-un-lft-identity 85.5%

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

      4. distribute-rgt-out 85.5%

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

      5. metadata-eval 85.5%

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

   6. Applied egg-rr 85.5%

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

      1. *-commutative 85.5%

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

      2. count-2 85.5%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \cdot x.re \]

      5. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{0}{\color{blue}{0}} \cdot x.re \]

      6. metadata-eval 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{0 \cdot 0}}{0} \cdot x.re \]

      7. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)} \cdot 0}{0} \cdot x.re \]

      8. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right) \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}}{0} \cdot x.re \]

      9. *-commutative 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}{0} \cdot x.re \]

      10. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}{0} \cdot x.re \]

      11. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)}{0} \cdot x.re \]

      12. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(x.im \cdot x.re\right)\right)}{0} \cdot x.re \]

      13. metadata-eval 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right)}{\color{blue}{0 \cdot 0}} \cdot x.re \]

      14. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right)}{\color{blue}{\left(x.re \cdot x.im - x.re \cdot x.im\right)} \cdot 0} \cdot x.re \]

      15. +-inverses 0.0%

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

      16. *-commutative 0.0%

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

      17. *-commutative 0.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(\frac{0}{0} \cdot \left(x.im \cdot x.re\right) + \frac{0}{0} \cdot \left(x.im \cdot x.re\right)\right)} \cdot x.re \]

   10. Simplified 60.8%

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

   11. Taylor expanded in x.im around 0 34.9%

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

   13. Simplified 35.8%

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

   if 3.39999999999999983e-77 < x.im

   1. Initial program 81.2%

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

      1. difference-of-squares 85.6%

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

      2. *-commutative 85.6%

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

   4. Applied egg-rr 85.6%

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

      1. expm1-log1p-u 47.2%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)\right)} \cdot x.re \]

      2. expm1-undefine 46.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)} - 1\right)} \cdot x.re \]

      3. *-commutative 46.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)} - 1\right) \cdot x.re \]

      4. flip-+ 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \cdot x.re \]

      5. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \cdot x.re \]

      6. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\frac{0}{\color{blue}{0}}\right)} - 1\right) \cdot x.re \]

   6. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0}{0}\right)} - 1\right)} \cdot x.re \]

   8. Simplified 91.0%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.4 \cdot 10^{-77}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) - x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 7 \cdot 10^{-25}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 7e-25)
    (* x.re (* x.im_m (+ x.re -1.0)))
    (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -1.0))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7e-25) {
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 7d-25) then
        tmp = x_46re * (x_46im_m * (x_46re + (-1.0d0)))
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-1.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7e-25) {
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 7e-25:
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0))
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 7e-25)
		tmp = Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re + -1.0)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -1.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 7e-25)
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0));
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -1.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 7e-25], N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.0000000000000004e-25

   1. Initial program 85.1%

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

      1. difference-of-squares 85.1%

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

      2. *-commutative 85.1%

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

   4. Applied egg-rr 85.1%

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

      1. *-commutative 85.1%

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

      2. *-un-lft-identity 85.1%

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

      3. *-un-lft-identity 85.1%

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

      4. distribute-rgt-out 85.1%

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

      5. metadata-eval 85.1%

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

   6. Applied egg-rr 85.1%

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

      1. *-commutative 85.1%

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

      2. count-2 85.1%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \cdot x.re \]

      5. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{0}{\color{blue}{0}} \cdot x.re \]

      6. metadata-eval 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{0 \cdot 0}}{0} \cdot x.re \]

      7. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)} \cdot 0}{0} \cdot x.re \]

      8. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right) \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}}{0} \cdot x.re \]

      9. *-commutative 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}{0} \cdot x.re \]

      10. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}{0} \cdot x.re \]

      11. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)}{0} \cdot x.re \]

      12. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(x.im \cdot x.re\right)\right)}{0} \cdot x.re \]

      13. metadata-eval 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right)}{\color{blue}{0 \cdot 0}} \cdot x.re \]

      14. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right)}{\color{blue}{\left(x.re \cdot x.im - x.re \cdot x.im\right)} \cdot 0} \cdot x.re \]

      15. +-inverses 0.0%

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

      16. *-commutative 0.0%

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

      17. *-commutative 0.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(\frac{0}{0} \cdot \left(x.im \cdot x.re\right) + \frac{0}{0} \cdot \left(x.im \cdot x.re\right)\right)} \cdot x.re \]

   10. Simplified 60.6%

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

   11. Taylor expanded in x.im around 0 35.1%

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

   13. Simplified 36.0%

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

   if 7.0000000000000004e-25 < x.im

   1. Initial program 82.2%

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

      1. difference-of-squares 87.0%

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

      2. *-commutative 87.0%

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

   4. Applied egg-rr 87.0%

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

      1. *-commutative 87.0%

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

      2. *-un-lft-identity 87.0%

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

      3. *-un-lft-identity 87.0%

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

      4. distribute-rgt-out 87.0%

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

      5. metadata-eval 87.0%

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

   6. Applied egg-rr 87.0%

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

      1. add-log-exp 48.7%

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

      2. *-commutative 48.7%

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

      3. add-log-exp 87.0%

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

      4. count-2 87.0%

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

      5. *-commutative 87.0%

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

      6. expm1-log1p-u 86.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

      7. expm1-undefine 86.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

      8. *-commutative 86.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

      9. *-commutative 86.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

      10. flip-+ 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

      11. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

      12. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

   10. Simplified 94.5%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7 \cdot 10^{-25}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 24.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 9.5 \cdot 10^{-38}:\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(x.im\_m \cdot \left(x.im\_m - x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 9.5e-38)
    (* (+ x.im_m x.re) (* x.im_m (- x.im_m x.re)))
    (* x.re (* x.im_m (+ x.re -1.0))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 9.5e-38) {
		tmp = (x_46_im_m + x_46_re) * (x_46_im_m * (x_46_im_m - x_46_re));
	} else {
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 9.5d-38) then
        tmp = (x_46im_m + x_46re) * (x_46im_m * (x_46im_m - x_46re))
    else
        tmp = x_46re * (x_46im_m * (x_46re + (-1.0d0)))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 9.5e-38) {
		tmp = (x_46_im_m + x_46_re) * (x_46_im_m * (x_46_im_m - x_46_re));
	} else {
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 9.5e-38:
		tmp = (x_46_im_m + x_46_re) * (x_46_im_m * (x_46_im_m - x_46_re))
	else:
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 9.5e-38)
		tmp = Float64(Float64(x_46_im_m + x_46_re) * Float64(x_46_im_m * Float64(x_46_im_m - x_46_re)));
	else
		tmp = Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re + -1.0)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 9.5e-38)
		tmp = (x_46_im_m + x_46_re) * (x_46_im_m * (x_46_im_m - x_46_re));
	else
		tmp = x_46_re * (x_46_im_m * (x_46_re + -1.0));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 9.5e-38], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$im$95$m - x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 9.5000000000000009e-38

   1. Initial program 90.8%

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

      1. difference-of-squares 91.4%

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

      2. *-commutative 91.4%

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

   4. Applied egg-rr 91.4%

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

      1. *-commutative 91.4%

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

      2. distribute-rgt-in 88.6%

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

      3. distribute-lft-in 84.2%

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

   6. Applied egg-rr 84.2%

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

   8. Simplified 88.6%

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

      1. +-commutative 88.6%

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

      2. distribute-rgt-in 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-*l* 84.2%

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

      5. pow2 84.2%

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

      6. *-commutative 84.2%

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

      7. associate-*l* 87.7%

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

      8. *-commutative 87.7%

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

   10. Applied egg-rr 87.7%

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

   12. Simplified 94.9%

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

       1. add-log-exp 60.9%

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

       2. +-commutative 60.9%

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

       3. exp-sum 52.0%

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

   14. Applied egg-rr 22.3%

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

   if 9.5000000000000009e-38 < x.re

   1. Initial program 68.8%

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

      1. difference-of-squares 71.5%

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

      2. *-commutative 71.5%

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

   4. Applied egg-rr 71.5%

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

      1. *-commutative 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. *-un-lft-identity 71.5%

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

      4. distribute-rgt-out 71.5%

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

      5. metadata-eval 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. *-commutative 71.5%

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

      2. count-2 71.5%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \cdot x.re \]

      5. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{0}{\color{blue}{0}} \cdot x.re \]

      6. metadata-eval 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{0 \cdot 0}}{0} \cdot x.re \]

      7. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)} \cdot 0}{0} \cdot x.re \]

      8. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right) \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}}{0} \cdot x.re \]

      9. *-commutative 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}{0} \cdot x.re \]

      10. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}{0} \cdot x.re \]

      11. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)}{0} \cdot x.re \]

      12. *-commutative 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(x.im \cdot x.re\right)\right)}{0} \cdot x.re \]

      13. metadata-eval 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right)}{\color{blue}{0 \cdot 0}} \cdot x.re \]

      14. +-inverses 0.0%

          \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right)}{\color{blue}{\left(x.re \cdot x.im - x.re \cdot x.im\right)} \cdot 0} \cdot x.re \]

      15. +-inverses 0.0%

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

      16. *-commutative 0.0%

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

      17. *-commutative 0.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(\frac{0}{0} \cdot \left(x.im \cdot x.re\right) + \frac{0}{0} \cdot \left(x.im \cdot x.re\right)\right)} \cdot x.re \]

   10. Simplified 55.6%

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

   11. Taylor expanded in x.im around 0 33.3%

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

   13. Simplified 34.8%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 9.5 \cdot 10^{-38}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.im - x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. difference-of-squares 85.5%

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

   2. *-commutative 85.5%

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

4. Applied egg-rr 85.5%

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

   1. *-commutative 85.5%

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

   2. *-un-lft-identity 85.5%

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

   3. *-un-lft-identity 85.5%

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

   4. distribute-rgt-out 85.5%

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

   5. metadata-eval 85.5%

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

6. Applied egg-rr 85.5%

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

   1. *-commutative 85.5%

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

   2. count-2 85.5%

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

   3. flip-+ 0.0%

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

   4. +-inverses 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \cdot x.re \]

   5. +-inverses 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{0}{\color{blue}{0}} \cdot x.re \]

   6. metadata-eval 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{0 \cdot 0}}{0} \cdot x.re \]

   7. +-inverses 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)} \cdot 0}{0} \cdot x.re \]

   8. +-inverses 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right) \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}}{0} \cdot x.re \]

   9. *-commutative 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}{0} \cdot x.re \]

   10. *-commutative 0.0%

       \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right)}{0} \cdot x.re \]

   11. *-commutative 0.0%

       \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)}{0} \cdot x.re \]

   12. *-commutative 0.0%

       \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(x.im \cdot x.re\right)\right)}{0} \cdot x.re \]

   13. metadata-eval 0.0%

       \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right)}{\color{blue}{0 \cdot 0}} \cdot x.re \]

   14. +-inverses 0.0%

       \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \frac{\left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\right)}{\color{blue}{\left(x.re \cdot x.im - x.re \cdot x.im\right)} \cdot 0} \cdot x.re \]

   15. +-inverses 0.0%

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

   16. *-commutative 0.0%

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

   17. *-commutative 0.0%

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

8. Applied egg-rr 0.0%

   \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{\left(\frac{0}{0} \cdot \left(x.im \cdot x.re\right) + \frac{0}{0} \cdot \left(x.im \cdot x.re\right)\right)} \cdot x.re \]

10. Simplified 66.5%

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

11. Taylor expanded in x.im around 0 32.5%

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

13. Simplified 33.2%

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

14. Final simplification 33.2%

    \[\leadsto x.re \cdot \left(x.im \cdot \left(x.re + -1\right)\right) \]
15. Add Preprocessing

Alternative 10: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot 9 \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m) :precision binary64 (* x.im_s 9.0))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * 9.0;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * 9.0d0
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * 9.0;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	return x_46_im_s * 9.0

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * 9.0)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * 9.0;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * 9.0), $MachinePrecision]

Derivation

1. Initial program 84.4%

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

3. Simplified 87.8%

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

5. Taylor expanded in x.im around -inf 60.6%

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

7. Simplified 6.4%

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

8. Taylor expanded in x.im around 0 2.7%

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

Alternative 11: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

3. Simplified 87.8%

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

5. Taylor expanded in x.re around 0 57.7%

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

7. Simplified 2.6%

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

Developer target: 91.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024086 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (+ (* (* x.re x.im) (* 2.0 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)))