math.cube on complex, real part

Percentage Accurate: 81.8% → 97.3%

Time: 6.2s

Alternatives: 7

Speedup: 0.8×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Initial Program: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \sqrt{x.re\_m \cdot 3}\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.6 \cdot 10^{+188}:\\ \;\;\;\;\left({x.re\_m}^{1.5} + t\_0\right) \cdot \left({x.re\_m}^{1.5} - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot -3\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (sqrt (* x.re_m 3.0)))))
   (*
    x.re_s
    (if (<= x.im_m 2.6e+188)
      (* (+ (pow x.re_m 1.5) t_0) (- (pow x.re_m 1.5) t_0))
      (* x.im_m (* x.im_m (* x.re_m -3.0)))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_im_m * sqrt((x_46_re_m * 3.0));
	double tmp;
	if (x_46_im_m <= 2.6e+188) {
		tmp = (pow(x_46_re_m, 1.5) + t_0) * (pow(x_46_re_m, 1.5) - t_0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im_m * sqrt((x_46re_m * 3.0d0))
    if (x_46im_m <= 2.6d+188) then
        tmp = ((x_46re_m ** 1.5d0) + t_0) * ((x_46re_m ** 1.5d0) - t_0)
    else
        tmp = x_46im_m * (x_46im_m * (x_46re_m * (-3.0d0)))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_im_m * Math.sqrt((x_46_re_m * 3.0));
	double tmp;
	if (x_46_im_m <= 2.6e+188) {
		tmp = (Math.pow(x_46_re_m, 1.5) + t_0) * (Math.pow(x_46_re_m, 1.5) - t_0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	t_0 = x_46_im_m * math.sqrt((x_46_re_m * 3.0))
	tmp = 0
	if x_46_im_m <= 2.6e+188:
		tmp = (math.pow(x_46_re_m, 1.5) + t_0) * (math.pow(x_46_re_m, 1.5) - t_0)
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_im_m * sqrt(Float64(x_46_re_m * 3.0)))
	tmp = 0.0
	if (x_46_im_m <= 2.6e+188)
		tmp = Float64(Float64((x_46_re_m ^ 1.5) + t_0) * Float64((x_46_re_m ^ 1.5) - t_0));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m * -3.0)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = x_46_im_m * sqrt((x_46_re_m * 3.0));
	tmp = 0.0;
	if (x_46_im_m <= 2.6e+188)
		tmp = ((x_46_re_m ^ 1.5) + t_0) * ((x_46_re_m ^ 1.5) - t_0);
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[Sqrt[N[(x$46$re$95$m * 3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 2.6e+188], N[(N[(N[Power[x$46$re$95$m, 1.5], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(N[Power[x$46$re$95$m, 1.5], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.59999999999999987e188

   1. Initial program 87.0%

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

   3. Simplified 85.1%

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

      1. sqr-pow 42.0%

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

      2. add-sqr-sqrt 42.0%

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

      3. difference-of-squares 45.0%

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

      4. metadata-eval 45.0%

         \[\leadsto \left({x.re}^{\color{blue}{1.5}} + \sqrt{x.re \cdot \left(3 \cdot \left(x.im \cdot x.im\right)\right)}\right) \cdot \left({x.re}^{\left(\frac{3}{2}\right)} - \sqrt{x.re \cdot \left(3 \cdot \left(x.im \cdot x.im\right)\right)}\right) \]

      5. associate-*r* 45.0%

         \[\leadsto \left({x.re}^{1.5} + \sqrt{\color{blue}{\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.im\right)}}\right) \cdot \left({x.re}^{\left(\frac{3}{2}\right)} - \sqrt{x.re \cdot \left(3 \cdot \left(x.im \cdot x.im\right)\right)}\right) \]

      6. sqrt-prod 45.0%

         \[\leadsto \left({x.re}^{1.5} + \color{blue}{\sqrt{x.re \cdot 3} \cdot \sqrt{x.im \cdot x.im}}\right) \cdot \left({x.re}^{\left(\frac{3}{2}\right)} - \sqrt{x.re \cdot \left(3 \cdot \left(x.im \cdot x.im\right)\right)}\right) \]

      7. sqrt-prod 23.3%

         \[\leadsto \left({x.re}^{1.5} + \sqrt{x.re \cdot 3} \cdot \color{blue}{\left(\sqrt{x.im} \cdot \sqrt{x.im}\right)}\right) \cdot \left({x.re}^{\left(\frac{3}{2}\right)} - \sqrt{x.re \cdot \left(3 \cdot \left(x.im \cdot x.im\right)\right)}\right) \]

      8. add-sqr-sqrt 37.0%

         \[\leadsto \left({x.re}^{1.5} + \sqrt{x.re \cdot 3} \cdot \color{blue}{x.im}\right) \cdot \left({x.re}^{\left(\frac{3}{2}\right)} - \sqrt{x.re \cdot \left(3 \cdot \left(x.im \cdot x.im\right)\right)}\right) \]

      9. metadata-eval 37.0%

         \[\leadsto \left({x.re}^{1.5} + \sqrt{x.re \cdot 3} \cdot x.im\right) \cdot \left({x.re}^{\color{blue}{1.5}} - \sqrt{x.re \cdot \left(3 \cdot \left(x.im \cdot x.im\right)\right)}\right) \]

      10. associate-*r* 37.0%

          \[\leadsto \left({x.re}^{1.5} + \sqrt{x.re \cdot 3} \cdot x.im\right) \cdot \left({x.re}^{1.5} - \sqrt{\color{blue}{\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.im\right)}}\right) \]

      11. sqrt-prod 38.7%

          \[\leadsto \left({x.re}^{1.5} + \sqrt{x.re \cdot 3} \cdot x.im\right) \cdot \left({x.re}^{1.5} - \color{blue}{\sqrt{x.re \cdot 3} \cdot \sqrt{x.im \cdot x.im}}\right) \]

      12. sqrt-prod 25.7%

          \[\leadsto \left({x.re}^{1.5} + \sqrt{x.re \cdot 3} \cdot x.im\right) \cdot \left({x.re}^{1.5} - \sqrt{x.re \cdot 3} \cdot \color{blue}{\left(\sqrt{x.im} \cdot \sqrt{x.im}\right)}\right) \]

      13. add-sqr-sqrt 50.6%

          \[\leadsto \left({x.re}^{1.5} + \sqrt{x.re \cdot 3} \cdot x.im\right) \cdot \left({x.re}^{1.5} - \sqrt{x.re \cdot 3} \cdot \color{blue}{x.im}\right) \]

   6. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{\left({x.re}^{1.5} + \sqrt{x.re \cdot 3} \cdot x.im\right) \cdot \left({x.re}^{1.5} - \sqrt{x.re \cdot 3} \cdot x.im\right)} \]

   if 2.59999999999999987e188 < x.im

   1. Initial program 45.7%

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

   3. Taylor expanded in x.re around 0 45.7%

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

   4. Taylor expanded in x.im around inf 75.7%

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

      1. *-commutative 75.7%

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

   6. Simplified 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*r* 75.7%

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

      3. unpow2 75.7%

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

      4. associate-*r* 85.0%

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

   8. Applied egg-rr 85.0%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.6 \cdot 10^{+188}:\\ \;\;\;\;\left({x.re}^{1.5} + x.im \cdot \sqrt{x.re \cdot 3}\right) \cdot \left({x.re}^{1.5} - x.im \cdot \sqrt{x.re \cdot 3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.35 \cdot 10^{+246}:\\ \;\;\;\;\left(x.im\_m + x.re\_m\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.35e+246)
    (-
     (* (+ x.im_m x.re_m) (* x.re_m (- x.re_m x.im_m)))
     (* x.im_m (+ (* x.im_m x.re_m) (* x.im_m x.re_m))))
    (pow x.re_m 3.0))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 2.35e+246) {
		tmp = ((x_46_im_m + x_46_re_m) * (x_46_re_m * (x_46_re_m - x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 2.35d+246) then
        tmp = ((x_46im_m + x_46re_m) * (x_46re_m * (x_46re_m - x_46im_m))) - (x_46im_m * ((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)))
    else
        tmp = x_46re_m ** 3.0d0
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 2.35e+246) {
		tmp = ((x_46_im_m + x_46_re_m) * (x_46_re_m * (x_46_re_m - x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	} else {
		tmp = Math.pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 2.35e+246:
		tmp = ((x_46_im_m + x_46_re_m) * (x_46_re_m * (x_46_re_m - x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))
	else:
		tmp = math.pow(x_46_re_m, 3.0)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 2.35e+246)
		tmp = Float64(Float64(Float64(x_46_im_m + x_46_re_m) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m))));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 2.35e+246)
		tmp = ((x_46_im_m + x_46_re_m) * (x_46_re_m * (x_46_re_m - x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.35e+246], N[(N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.34999999999999988e246

   1. Initial program 85.4%

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

      1. difference-of-squares 89.5%

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

      2. associate-*l* 96.8%

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

   4. Applied egg-rr 96.8%

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

   if 2.34999999999999988e246 < x.re

   1. Initial program 53.8%

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

      1. difference-of-squares 61.5%

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

      2. associate-*l* 61.5%

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

   4. Applied egg-rr 61.5%

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

   5. Taylor expanded in x.re around inf 92.3%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.35 \cdot 10^{+246}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5.2 \cdot 10^{+247}:\\ \;\;\;\;\left(x.im\_m + x.re\_m\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m + x.re\_m\right) \cdot \left(x.im\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5.2e+247)
    (-
     (* (+ x.im_m x.re_m) (* x.re_m (- x.re_m x.im_m)))
     (* x.im_m (+ (* x.im_m x.re_m) (* x.im_m x.re_m))))
    (* (+ x.im_m x.re_m) (* x.im_m x.re_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 5.2e+247) {
		tmp = ((x_46_im_m + x_46_re_m) * (x_46_re_m * (x_46_re_m - x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	} else {
		tmp = (x_46_im_m + x_46_re_m) * (x_46_im_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 5.2d+247) then
        tmp = ((x_46im_m + x_46re_m) * (x_46re_m * (x_46re_m - x_46im_m))) - (x_46im_m * ((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)))
    else
        tmp = (x_46im_m + x_46re_m) * (x_46im_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 5.2e+247) {
		tmp = ((x_46_im_m + x_46_re_m) * (x_46_re_m * (x_46_re_m - x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	} else {
		tmp = (x_46_im_m + x_46_re_m) * (x_46_im_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 5.2e+247:
		tmp = ((x_46_im_m + x_46_re_m) * (x_46_re_m * (x_46_re_m - x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))
	else:
		tmp = (x_46_im_m + x_46_re_m) * (x_46_im_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 5.2e+247)
		tmp = Float64(Float64(Float64(x_46_im_m + x_46_re_m) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m))));
	else
		tmp = Float64(Float64(x_46_im_m + x_46_re_m) * Float64(x_46_im_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 5.2e+247)
		tmp = ((x_46_im_m + x_46_re_m) * (x_46_re_m * (x_46_re_m - x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	else
		tmp = (x_46_im_m + x_46_re_m) * (x_46_im_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 5.2e+247], N[(N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.19999999999999981e247

   1. Initial program 85.4%

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

      1. difference-of-squares 89.5%

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

      2. associate-*l* 96.8%

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

   4. Applied egg-rr 96.8%

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

   if 5.19999999999999981e247 < x.re

   1. Initial program 50.0%

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

      1. difference-of-squares 58.3%

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

      2. associate-*l* 58.3%

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

   4. Applied egg-rr 58.3%

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

   5. Taylor expanded in x.re around 0 41.7%

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

      1. mul-1-neg 41.7%

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

      2. *-commutative 41.7%

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

      3. distribute-rgt-neg-in 41.7%

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

   7. Simplified 41.7%

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

      1. add-sqr-sqrt 41.7%

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

      2. sqrt-unprod 58.3%

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

      3. distribute-rgt-neg-out 58.3%

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

      4. distribute-rgt-neg-out 58.3%

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

      5. sqr-neg 58.3%

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

      6. sqrt-unprod 16.7%

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

      7. add-sqr-sqrt 16.7%

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

      8. add-log-exp 0.0%

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

      9. *-commutative 0.0%

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

      10. fma-define 0.0%

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

      11. *-commutative 0.0%

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

      12. exp-prod 16.7%

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

      13. add-sqr-sqrt 16.7%

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

      14. sqrt-unprod 16.7%

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

      15. sqr-neg 16.7%

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

      16. distribute-rgt-neg-out 16.7%

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

      17. distribute-rgt-neg-out 16.7%

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

   9. Applied egg-rr 50.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5.2 \cdot 10^{+247}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 90000:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \left(x.im\_m + x.re\_m\right)\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 90000.0)
    (-
     (* x.im_m (* x.re_m (+ x.im_m x.re_m)))
     (* x.im_m (+ (* x.im_m x.re_m) (* x.im_m x.re_m))))
    (* x.im_m (* x.im_m (* x.re_m -3.0))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 90000.0) {
		tmp = (x_46_im_m * (x_46_re_m * (x_46_im_m + x_46_re_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 90000.0d0) then
        tmp = (x_46im_m * (x_46re_m * (x_46im_m + x_46re_m))) - (x_46im_m * ((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)))
    else
        tmp = x_46im_m * (x_46im_m * (x_46re_m * (-3.0d0)))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 90000.0) {
		tmp = (x_46_im_m * (x_46_re_m * (x_46_im_m + x_46_re_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 90000.0:
		tmp = (x_46_im_m * (x_46_re_m * (x_46_im_m + x_46_re_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 90000.0)
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_im_m + x_46_re_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m))));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m * -3.0)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 90000.0)
		tmp = (x_46_im_m * (x_46_re_m * (x_46_im_m + x_46_re_m))) - (x_46_im_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 90000.0], N[(N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 9e4

   1. Initial program 88.8%

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

      1. difference-of-squares 91.3%

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

      2. associate-*l* 97.1%

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

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in x.re around 0 55.1%

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

      1. mul-1-neg 55.1%

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

      2. *-commutative 55.1%

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

      3. distribute-rgt-neg-in 55.1%

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

   7. Simplified 55.1%

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

      1. associate-*r* 58.4%

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

      2. *-commutative 58.4%

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

      3. add-sqr-sqrt 40.9%

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

      4. sqrt-unprod 44.9%

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

      5. sqr-neg 44.9%

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

      6. sqrt-prod 12.9%

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

      7. add-sqr-sqrt 26.0%

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

   9. Applied egg-rr 26.0%

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

   if 9e4 < x.im

   1. Initial program 68.2%

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

   3. Taylor expanded in x.re around 0 73.1%

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

   4. Taylor expanded in x.im around inf 73.1%

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

      1. *-commutative 73.1%

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

   6. Simplified 73.1%

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

      1. *-commutative 73.1%

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

      2. associate-*r* 73.0%

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

      3. unpow2 73.0%

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

      4. associate-*r* 83.6%

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

   8. Applied egg-rr 83.6%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 90000:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im + x.re\right)\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.7 \cdot 10^{+183}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m + x.re\_m\right) \cdot \left(x.im\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.7e+183)
    (* x.im_m (* x.im_m (* x.re_m -3.0)))
    (* (+ x.im_m x.re_m) (* x.im_m x.re_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 2.7e+183) {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0));
	} else {
		tmp = (x_46_im_m + x_46_re_m) * (x_46_im_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 2.7d+183) then
        tmp = x_46im_m * (x_46im_m * (x_46re_m * (-3.0d0)))
    else
        tmp = (x_46im_m + x_46re_m) * (x_46im_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 2.7e+183) {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0));
	} else {
		tmp = (x_46_im_m + x_46_re_m) * (x_46_im_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 2.7e+183:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0))
	else:
		tmp = (x_46_im_m + x_46_re_m) * (x_46_im_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 2.7e+183)
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m * -3.0)));
	else
		tmp = Float64(Float64(x_46_im_m + x_46_re_m) * Float64(x_46_im_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 2.7e+183)
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0));
	else
		tmp = (x_46_im_m + x_46_re_m) * (x_46_im_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.7e+183], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.69999999999999982e183

   1. Initial program 86.8%

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

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

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

   4. Taylor expanded in x.im around inf 52.1%

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

      1. *-commutative 52.1%

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

   6. Simplified 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-*r* 52.0%

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

      3. unpow2 52.0%

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

      4. associate-*r* 59.7%

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

   8. Applied egg-rr 59.7%

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

   if 2.69999999999999982e183 < x.re

   1. Initial program 56.0%

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

      1. difference-of-squares 80.0%

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

      2. associate-*l* 80.0%

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

   4. Applied egg-rr 80.0%

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

   5. Taylor expanded in x.re around 0 56.2%

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

      1. mul-1-neg 56.2%

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

      2. *-commutative 56.2%

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

      3. distribute-rgt-neg-in 56.2%

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

   7. Simplified 56.2%

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

      1. add-sqr-sqrt 48.2%

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

      2. sqrt-unprod 60.8%

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

      3. distribute-rgt-neg-out 60.8%

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

      4. distribute-rgt-neg-out 60.8%

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

      5. sqr-neg 60.8%

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

      6. sqrt-unprod 12.6%

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

      7. add-sqr-sqrt 12.6%

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

      8. add-log-exp 0.6%

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

      9. *-commutative 0.6%

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

      10. fma-define 0.6%

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

      11. *-commutative 0.6%

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

      12. exp-prod 8.6%

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

      13. add-sqr-sqrt 8.6%

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

      14. sqrt-unprod 8.6%

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

      15. sqr-neg 8.6%

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

      16. distribute-rgt-neg-out 8.6%

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

      17. distribute-rgt-neg-out 8.6%

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

   9. Applied egg-rr 28.6%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.7 \cdot 10^{+183}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0)));
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * (x_46im_m * (x_46im_m * (x_46re_m * (-3.0d0))))
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0)));
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * (x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0)))

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m * -3.0))))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * (x_46_im_m * (x_46_im_m * (x_46_re_m * -3.0)));
end

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in x.re around 0 86.1%

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

4. Taylor expanded in x.im around inf 49.4%

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

   1. *-commutative 49.4%

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

6. Simplified 49.4%

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

   1. *-commutative 49.4%

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

   2. associate-*r* 49.4%

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

   3. unpow2 49.4%

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

   4. associate-*r* 56.3%

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

8. Applied egg-rr 56.3%

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

9. Final simplification 56.3%

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

Alternative 7: 50.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_re_m * (x_46_im_m * (x_46_im_m * -3.0)));
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * (x_46re_m * (x_46im_m * (x_46im_m * (-3.0d0))))
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_re_m * (x_46_im_m * (x_46_im_m * -3.0)));
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * (x_46_re_m * (x_46_im_m * (x_46_im_m * -3.0)))

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_im_m * -3.0))))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * (x_46_re_m * (x_46_im_m * (x_46_im_m * -3.0)));
end

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in x.re around 0 86.1%

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

4. Taylor expanded in x.im around inf 49.4%

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

   1. *-commutative 49.4%

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

6. Simplified 49.4%

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

   1. *-commutative 49.4%

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

   2. unpow2 49.4%

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

   3. associate-*r* 49.3%

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

8. Applied egg-rr 49.3%

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

9. Final simplification 49.3%

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

Developer target: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * 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_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024096 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))