math.cube on complex, imaginary part

Percentage Accurate: 82.8% → 99.8%

Time: 7.2s

Alternatives: 6

Speedup: 0.8×

• 44.7% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+96}:\\ \;\;\;\;\left(x.re \cdot \left(x.im\_m \cdot x.re\right)\right) \cdot 3 - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right)\\ \end{array} \end{array} \]

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+96)
    (- (* (* 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))))))

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+96) {
		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));
	}
	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+96) 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))
    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+96) {
		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));
	}
	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+96:
		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))
	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+96)
		tmp = Float64(Float64(Float64(x_46_re * Float64(x_46_im_m * x_46_re)) * 3.0) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + 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 <= 5e+96)
		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));
	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+96], N[(N[(N[(x$46$re * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000004e96

   1. Initial program 89.6%

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

   3. Simplified 92.9%

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

   if 5.0000000000000004e96 < x.im

   1. Initial program 71.8%

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

      1. difference-of-squares 84.6%

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

      2. *-commutative 84.6%

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

   4. Applied egg-rr 84.6%

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

      1. expm1-log1p-u 84.6%

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

      2. expm1-undefine 84.6%

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

      3. *-commutative 84.6%

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

      4. *-commutative 84.6%

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

      5. count-2 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      6. *-commutative 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      7. associate-*r* 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 2\right)\right)}\right)} - 1\right) \]

      8. associate-*r* 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      9. *-commutative 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      10. count-2 84.6%

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

      11. flip-+ 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 100.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2 \cdot 10^{+93}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re \cdot 3\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right)\\ \end{array} \end{array} \]

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 2e+93)
    (- (* 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))))))

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 <= 2e+93) {
		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));
	}
	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 <= 2d+93) 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))
    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 <= 2e+93) {
		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));
	}
	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 <= 2e+93:
		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))
	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 <= 2e+93)
		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(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + 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 <= 2e+93)
		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));
	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, 2e+93], 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[(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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.00000000000000009e93

   1. Initial program 89.6%

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

   3. Simplified 92.9%

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

   if 2.00000000000000009e93 < x.im

   1. Initial program 71.8%

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

      1. difference-of-squares 84.6%

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

      2. *-commutative 84.6%

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

   4. Applied egg-rr 84.6%

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

      1. expm1-log1p-u 84.6%

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

      2. expm1-undefine 84.6%

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

      3. *-commutative 84.6%

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

      4. *-commutative 84.6%

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

      5. count-2 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      6. *-commutative 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      7. associate-*r* 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 2\right)\right)}\right)} - 1\right) \]

      8. associate-*r* 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      9. *-commutative 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      10. count-2 84.6%

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

      11. flip-+ 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 100.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2 \cdot 10^{+93}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2 \cdot 10^{+93}:\\ \;\;\;\;x.re \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot 3\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right)\\ \end{array} \end{array} \]

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 2e+93)
    (- (* 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))))))

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 <= 2e+93) {
		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));
	}
	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 <= 2d+93) 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))
    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 <= 2e+93) {
		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));
	}
	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 <= 2e+93:
		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))
	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 <= 2e+93)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) * 3.0)) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + 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 <= 2e+93)
		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));
	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, 2e+93], N[(N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.00000000000000009e93

   1. Initial program 89.6%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x.im around 0 92.9%

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

   if 2.00000000000000009e93 < x.im

   1. Initial program 71.8%

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

      1. difference-of-squares 84.6%

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

      2. *-commutative 84.6%

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

   4. Applied egg-rr 84.6%

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

      1. expm1-log1p-u 84.6%

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

      2. expm1-undefine 84.6%

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

      3. *-commutative 84.6%

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

      4. *-commutative 84.6%

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

      5. count-2 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      6. *-commutative 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      7. associate-*r* 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 2\right)\right)}\right)} - 1\right) \]

      8. associate-*r* 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      9. *-commutative 84.6%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      10. count-2 84.6%

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

      11. flip-+ 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 100.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2 \cdot 10^{+93}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{+291}:\\ \;\;\;\;t\_0 + x.re \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im\_m \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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))))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (*
    x.im_s
    (if (<= t_1 1e+291)
      (+ t_0 (* x.re (* (* x.im_m x.re) 2.0)))
      (if (<= t_1 INFINITY) (* x.re (* x.re (* x.im_m 3.0))) t_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 t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= 1e+291) {
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_re * (x_46_re * (x_46_im_m * 3.0));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= 1e+291) {
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re * (x_46_re * (x_46_im_m * 3.0));
	} else {
		tmp = t_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))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))
	tmp = 0
	if t_1 <= 1e+291:
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0))
	elif t_1 <= math.inf:
		tmp = x_46_re * (x_46_re * (x_46_im_m * 3.0))
	else:
		tmp = t_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)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_1 <= 1e+291)
		tmp = Float64(t_0 + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im_m * 3.0)));
	else
		tmp = t_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));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	tmp = 0.0;
	if (t_1 <= 1e+291)
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	elseif (t_1 <= Inf)
		tmp = x_46_re * (x_46_re * (x_46_im_m * 3.0));
	else
		tmp = t_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]}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, 1e+291], N[(t$95$0 + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x$46$re * N[(x$46$re * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 9.9999999999999996e290

   1. Initial program 97.3%

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

      1. difference-of-squares 97.3%

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

      2. *-commutative 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. *-commutative 97.3%

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

      2. *-un-lft-identity 97.3%

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

      3. *-un-lft-identity 97.3%

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

      4. distribute-rgt-out 97.3%

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

      5. metadata-eval 97.3%

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

   6. Applied egg-rr 97.3%

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

   if 9.9999999999999996e290 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 90.6%

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

   3. Taylor expanded in x.im around 0 44.2%

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

   5. Simplified 44.2%

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

      1. add-sqr-sqrt 44.0%

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

      2. pow2 44.0%

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

      3. associate-*r* 44.0%

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

      4. sqrt-prod 43.7%

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

      5. sqrt-pow1 53.0%

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

      6. metadata-eval 53.0%

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

      7. pow1 53.0%

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

   7. Applied egg-rr 53.0%

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

      1. unpow2 53.0%

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

      2. swap-sqr 43.7%

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

      3. add-sqr-sqrt 44.2%

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

      4. associate-*r* 53.4%

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

   9. Applied egg-rr 53.4%

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

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

   1. Initial program 0.0%

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

      1. difference-of-squares 41.7%

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

      2. *-commutative 41.7%

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

   4. Applied egg-rr 41.7%

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

      1. expm1-log1p-u 20.8%

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

      2. expm1-undefine 20.8%

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

      3. *-commutative 20.8%

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

      4. *-commutative 20.8%

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

      5. count-2 20.8%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      6. *-commutative 20.8%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      7. associate-*r* 20.8%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 2\right)\right)}\right)} - 1\right) \]

      8. associate-*r* 20.8%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      9. *-commutative 20.8%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      10. count-2 20.8%

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

      11. flip-+ 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 100.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 10^{+291}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.2 \cdot 10^{-71} \lor \neg \left(x.im\_m \leq 3.5 \cdot 10^{-52}\right) \land x.im\_m \leq 5.9 \cdot 10^{-34}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im\_m \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right)\\ \end{array} \end{array} \]

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 (or (<= x.im_m 3.2e-71)
          (and (not (<= x.im_m 3.5e-52)) (<= x.im_m 5.9e-34)))
    (* x.re (* x.re (* x.im_m 3.0)))
    (* x.im_m (* (- x.re x.im_m) (+ x.im_m 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 <= 3.2e-71) || (!(x_46_im_m <= 3.5e-52) && (x_46_im_m <= 5.9e-34))) {
		tmp = x_46_re * (x_46_re * (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));
	}
	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 <= 3.2d-71) .or. (.not. (x_46im_m <= 3.5d-52)) .and. (x_46im_m <= 5.9d-34)) then
        tmp = x_46re * (x_46re * (x_46im_m * 3.0d0))
    else
        tmp = x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + 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 <= 3.2e-71) || (!(x_46_im_m <= 3.5e-52) && (x_46_im_m <= 5.9e-34))) {
		tmp = x_46_re * (x_46_re * (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));
	}
	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 <= 3.2e-71) or (not (x_46_im_m <= 3.5e-52) and (x_46_im_m <= 5.9e-34)):
		tmp = x_46_re * (x_46_re * (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))
	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 <= 3.2e-71) || (!(x_46_im_m <= 3.5e-52) && (x_46_im_m <= 5.9e-34)))
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im_m * 3.0)));
	else
		tmp = Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + 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 <= 3.2e-71) || (~((x_46_im_m <= 3.5e-52)) && (x_46_im_m <= 5.9e-34)))
		tmp = x_46_re * (x_46_re * (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));
	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[Or[LessEqual[x$46$im$95$m, 3.2e-71], And[N[Not[LessEqual[x$46$im$95$m, 3.5e-52]], $MachinePrecision], LessEqual[x$46$im$95$m, 5.9e-34]]], N[(x$46$re * N[(x$46$re * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.1999999999999999e-71 or 3.5e-52 < x.im < 5.9000000000000002e-34

   1. Initial program 87.9%

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

   3. Taylor expanded in x.im around 0 66.9%

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

   5. Simplified 66.9%

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

      1. add-sqr-sqrt 40.3%

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

      2. pow2 40.3%

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

      3. associate-*r* 40.3%

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

      4. sqrt-prod 29.6%

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

      5. sqrt-pow1 32.1%

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

      6. metadata-eval 32.1%

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

      7. pow1 32.1%

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

   7. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. swap-sqr 29.6%

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

      3. add-sqr-sqrt 66.9%

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

      4. associate-*r* 71.8%

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

   9. Applied egg-rr 71.8%

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

   if 3.1999999999999999e-71 < x.im < 3.5e-52 or 5.9000000000000002e-34 < x.im

   1. Initial program 84.2%

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

      1. difference-of-squares 91.3%

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

      2. *-commutative 91.3%

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

   4. Applied egg-rr 91.3%

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

      1. expm1-log1p-u 90.7%

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

      2. expm1-undefine 89.5%

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

      3. *-commutative 89.5%

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

      4. *-commutative 89.5%

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

      5. count-2 89.5%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      6. *-commutative 89.5%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      7. associate-*r* 89.5%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 2\right)\right)}\right)} - 1\right) \]

      8. associate-*r* 89.5%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      9. *-commutative 89.5%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      10. count-2 89.5%

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

      11. flip-+ 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 92.9%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.2 \cdot 10^{-71} \lor \neg \left(x.im \leq 3.5 \cdot 10^{-52}\right) \land x.im \leq 5.9 \cdot 10^{-34}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in x.im around 0 58.7%

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

5. Simplified 58.7%

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

   1. add-sqr-sqrt 39.4%

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

   2. pow2 39.4%

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

   3. associate-*r* 39.4%

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

   4. sqrt-prod 31.6%

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

   5. sqrt-pow1 33.4%

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

   6. metadata-eval 33.4%

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

   7. pow1 33.4%

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

7. Applied egg-rr 33.4%

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

   1. unpow2 33.4%

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

   2. swap-sqr 31.6%

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

   3. add-sqr-sqrt 58.7%

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

   4. associate-*r* 62.3%

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

9. Applied egg-rr 62.3%

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

10. Final simplification 62.3%

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

Developer target: 91.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024097 
(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)))