math.cube on complex, real part

Percentage Accurate: 83.1% → 96.4%

Time: 7.7s

Alternatives: 12

Speedup: 0.5×

• 44.1% 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: 83.1% 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: 96.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 7.2 \cdot 10^{+84}:\\ \;\;\;\;{x.re\_m}^{3} + {\left(x.im \cdot \sqrt{x.re\_m}\right)}^{2} \cdot -3\\ \mathbf{elif}\;x.re\_m \leq 2.35 \cdot 10^{+148}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 7.2e+84)
    (+ (pow x.re_m 3.0) (* (pow (* x.im (sqrt x.re_m)) 2.0) -3.0))
    (if (<= x.re_m 2.35e+148)
      (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
      (* x.re_m (* x.re_m x.re_m))))))

C

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) {
	double tmp;
	if (x_46_re_m <= 7.2e+84) {
		tmp = pow(x_46_re_m, 3.0) + (pow((x_46_im * sqrt(x_46_re_m)), 2.0) * -3.0);
	} else if (x_46_re_m <= 2.35e+148) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 7.2d+84) then
        tmp = (x_46re_m ** 3.0d0) + (((x_46im * sqrt(x_46re_m)) ** 2.0d0) * (-3.0d0))
    else if (x_46re_m <= 2.35d+148) then
        tmp = x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double tmp;
	if (x_46_re_m <= 7.2e+84) {
		tmp = Math.pow(x_46_re_m, 3.0) + (Math.pow((x_46_im * Math.sqrt(x_46_re_m)), 2.0) * -3.0);
	} else if (x_46_re_m <= 2.35e+148) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

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):
	tmp = 0
	if x_46_re_m <= 7.2e+84:
		tmp = math.pow(x_46_re_m, 3.0) + (math.pow((x_46_im * math.sqrt(x_46_re_m)), 2.0) * -3.0)
	elif x_46_re_m <= 2.35e+148:
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

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)
	tmp = 0.0
	if (x_46_re_m <= 7.2e+84)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64((Float64(x_46_im * sqrt(x_46_re_m)) ^ 2.0) * -3.0));
	elseif (x_46_re_m <= 2.35e+148)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	tmp = 0.0;
	if (x_46_re_m <= 7.2e+84)
		tmp = (x_46_re_m ^ 3.0) + (((x_46_im * sqrt(x_46_re_m)) ^ 2.0) * -3.0);
	elseif (x_46_re_m <= 2.35e+148)
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 7.2e+84], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(N[Power[N[(x$46$im * N[Sqrt[x$46$re$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 2.35e+148], N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 7.1999999999999999e84

   1. Initial program 86.2%

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

   3. Simplified 85.8%

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

      1. add-sqr-sqrt 51.6%

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

      2. pow2 51.6%

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

      3. *-commutative 51.6%

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

      4. sqrt-prod 36.4%

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

      5. sqrt-prod 22.0%

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

      6. add-sqr-sqrt 40.4%

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

   6. Applied egg-rr 40.4%

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

   if 7.1999999999999999e84 < x.re < 2.3499999999999999e148

   1. Initial program 93.3%

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

      1. *-commutative 93.3%

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

      2. *-un-lft-identity 93.3%

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

      3. *-un-lft-identity 93.3%

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

      4. distribute-rgt-out 93.3%

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

      5. metadata-eval 93.3%

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

   4. Applied egg-rr 93.3%

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

      1. pow1 93.3%

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

      2. *-commutative 93.3%

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

      3. *-commutative 93.3%

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

      4. count-2 93.3%

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

      5. flip-+ 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x.im around 0 100.0%

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

   if 2.3499999999999999e148 < x.re

   1. Initial program 44.8%

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

   3. Simplified 44.8%

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

      1. cube-mult 44.8%

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

      2. associate-*l* 44.8%

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

      3. distribute-lft-out 72.4%

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

      4. pow2 72.4%

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

      5. pow2 72.4%

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

   6. Applied egg-rr 72.4%

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

   8. Applied egg-rr 86.2%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7.2 \cdot 10^{+84}:\\ \;\;\;\;{x.re}^{3} + {\left(x.im \cdot \sqrt{x.re}\right)}^{2} \cdot -3\\ \mathbf{elif}\;x.re \leq 2.35 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 10^{-218}:\\ \;\;\;\;t\_0 - x.im \cdot \left(2 \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (let* ((t_0 (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))))
   (*
    x.re_s
    (if (<= (- t_0 (* x.im (+ (* x.re_m x.im) (* x.re_m x.im)))) 1e-218)
      (- t_0 (* x.im (* 2.0 (* x.re_m x.im))))
      (pow x.re_m 3.0)))))

C

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) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double tmp;
	if ((t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e-218) {
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))
    if ((t_0 - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= 1d-218) then
        tmp = t_0 - (x_46im * (2.0d0 * (x_46re_m * x_46im)))
    else
        tmp = x_46re_m ** 3.0d0
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double tmp;
	if ((t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e-218) {
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)));
	} else {
		tmp = Math.pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Python

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):
	t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))
	tmp = 0
	if (t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e-218:
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)))
	else:
		tmp = math.pow(x_46_re_m, 3.0)
	return x_46_re_s * tmp

Julia

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)
	t_0 = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 1e-218)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(2.0 * Float64(x_46_re_m * x_46_im))));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	tmp = 0.0;
	if ((t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e-218)
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-218], N[(t$95$0 - N[(x$46$im * N[(2.0 * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1e-218

   1. Initial program 93.8%

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

      1. *-commutative 93.8%

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

      2. *-un-lft-identity 93.8%

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

      3. *-un-lft-identity 93.8%

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

      4. distribute-rgt-out 93.8%

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

      5. metadata-eval 93.8%

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

   4. Applied egg-rr 93.8%

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

   if 1e-218 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 67.6%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in x.re around inf 56.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-218}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(2 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 10^{-218}:\\ \;\;\;\;t\_0 - x.im \cdot \left(2 \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (let* ((t_0 (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))))
   (*
    x.re_s
    (if (<= (- t_0 (* x.im (+ (* x.re_m x.im) (* x.re_m x.im)))) 1e-218)
      (- t_0 (* x.im (* 2.0 (* x.re_m x.im))))
      (* x.re_m (* x.re_m x.re_m))))))

C

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) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double tmp;
	if ((t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e-218) {
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))
    if ((t_0 - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= 1d-218) then
        tmp = t_0 - (x_46im * (2.0d0 * (x_46re_m * x_46im)))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double tmp;
	if ((t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e-218) {
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

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):
	t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))
	tmp = 0
	if (t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e-218:
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)))
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

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)
	t_0 = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 1e-218)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(2.0 * Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	tmp = 0.0;
	if ((t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e-218)
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)));
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-218], N[(t$95$0 - N[(x$46$im * N[(2.0 * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1e-218

   1. Initial program 93.8%

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

      1. *-commutative 93.8%

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

      2. *-un-lft-identity 93.8%

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

      3. *-un-lft-identity 93.8%

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

      4. distribute-rgt-out 93.8%

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

      5. metadata-eval 93.8%

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

   4. Applied egg-rr 93.8%

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

   if 1e-218 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 67.6%

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

   3. Simplified 66.7%

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

      1. cube-mult 66.7%

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

      2. associate-*l* 66.7%

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

      3. distribute-lft-out 83.1%

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

      4. pow2 83.1%

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

      5. pow2 83.1%

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

   6. Applied egg-rr 83.1%

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

   8. Applied egg-rr 56.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-218}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(2 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 1.68 \cdot 10^{+146}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot -8 + \left(x.re\_m \cdot x.im\right) \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 1.68e+146)
    (* x.re_m (* x.re_m x.re_m))
    (* x.im (+ (* x.re_m -8.0) (* (* x.re_m x.im) -2.0))))))

C

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) {
	double tmp;
	if (x_46_im <= 1.68e+146) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = x_46_im * ((x_46_re_m * -8.0) + ((x_46_re_m * x_46_im) * -2.0));
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 1.68d+146) then
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    else
        tmp = x_46im * ((x_46re_m * (-8.0d0)) + ((x_46re_m * x_46im) * (-2.0d0)))
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double tmp;
	if (x_46_im <= 1.68e+146) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = x_46_im * ((x_46_re_m * -8.0) + ((x_46_re_m * x_46_im) * -2.0));
	}
	return x_46_re_s * tmp;
}

Python

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):
	tmp = 0
	if x_46_im <= 1.68e+146:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	else:
		tmp = x_46_im * ((x_46_re_m * -8.0) + ((x_46_re_m * x_46_im) * -2.0))
	return x_46_re_s * tmp

Julia

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)
	tmp = 0.0
	if (x_46_im <= 1.68e+146)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re_m * -8.0) + Float64(Float64(x_46_re_m * x_46_im) * -2.0)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	tmp = 0.0;
	if (x_46_im <= 1.68e+146)
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	else
		tmp = x_46_im * ((x_46_re_m * -8.0) + ((x_46_re_m * x_46_im) * -2.0));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 1.68e+146], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re$95$m * -8.0), $MachinePrecision] + N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.67999999999999992e146

   1. Initial program 86.2%

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

   3. Simplified 85.3%

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

      1. cube-mult 85.3%

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

      2. associate-*l* 85.2%

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

      3. distribute-lft-out 94.5%

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

      4. pow2 94.5%

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

      5. pow2 94.5%

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

   6. Applied egg-rr 94.5%

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

   8. Applied egg-rr 67.1%

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

   if 1.67999999999999992e146 < x.im

   1. Initial program 58.1%

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

      1. difference-of-squares 63.3%

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

   4. Applied egg-rr 63.3%

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

   6. Simplified 47.7%

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

   7. Taylor expanded in x.re around 0 63.1%

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

      1. *-commutative 63.1%

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

   9. Simplified 63.1%

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

   10. Taylor expanded in x.im around 0 63.1%

       \[\leadsto \color{blue}{x.im \cdot \left(-8 \cdot x.re + -2 \cdot \left(x.im \cdot x.re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.68 \cdot 10^{+146}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -8 + \left(x.re \cdot x.im\right) \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.35 \cdot 10^{+148}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.35e+148)
    (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
    (* x.re_m (* x.re_m x.re_m)))))

C

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) {
	double tmp;
	if (x_46_re_m <= 2.35e+148) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2.35d+148) then
        tmp = x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double tmp;
	if (x_46_re_m <= 2.35e+148) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

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):
	tmp = 0
	if x_46_re_m <= 2.35e+148:
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

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)
	tmp = 0.0
	if (x_46_re_m <= 2.35e+148)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	tmp = 0.0;
	if (x_46_re_m <= 2.35e+148)
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.35e+148], N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.3499999999999999e148

   1. Initial program 86.7%

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

      1. *-commutative 86.7%

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

      2. *-un-lft-identity 86.7%

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

      3. *-un-lft-identity 86.7%

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

      4. distribute-rgt-out 86.7%

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

      5. metadata-eval 86.7%

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

   4. Applied egg-rr 86.7%

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

      1. pow1 86.7%

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

      2. *-commutative 86.7%

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

      3. *-commutative 86.7%

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

      4. count-2 86.7%

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

      5. flip-+ 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 72.2%

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

   9. Taylor expanded in x.im around 0 72.2%

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

   if 2.3499999999999999e148 < x.re

   1. Initial program 44.8%

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

   3. Simplified 44.8%

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

      1. cube-mult 44.8%

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

      2. associate-*l* 44.8%

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

      3. distribute-lft-out 72.4%

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

      4. pow2 72.4%

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

      5. pow2 72.4%

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

   6. Applied egg-rr 72.4%

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

   8. Applied egg-rr 86.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.35 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 3.8 \cdot 10^{+228}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 3.8e+228)
    (* x.re_m (* x.re_m x.re_m))
    (* x.re_m (* x.re_m -27.0)))))

C

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) {
	double tmp;
	if (x_46_im <= 3.8e+228) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = x_46_re_m * (x_46_re_m * -27.0);
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 3.8d+228) then
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    else
        tmp = x_46re_m * (x_46re_m * (-27.0d0))
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double tmp;
	if (x_46_im <= 3.8e+228) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = x_46_re_m * (x_46_re_m * -27.0);
	}
	return x_46_re_s * tmp;
}

Python

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):
	tmp = 0
	if x_46_im <= 3.8e+228:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	else:
		tmp = x_46_re_m * (x_46_re_m * -27.0)
	return x_46_re_s * tmp

Julia

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)
	tmp = 0.0
	if (x_46_im <= 3.8e+228)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * -27.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	tmp = 0.0;
	if (x_46_im <= 3.8e+228)
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	else
		tmp = x_46_re_m * (x_46_re_m * -27.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 3.8e+228], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * -27.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.8000000000000002e228

   1. Initial program 83.0%

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

   3. Simplified 81.3%

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

      1. cube-mult 81.3%

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

      2. associate-*l* 81.3%

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

      3. distribute-lft-out 90.5%

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

      4. pow2 90.5%

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

      5. pow2 90.5%

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

   6. Applied egg-rr 90.5%

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

   8. Applied egg-rr 62.9%

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

   if 3.8000000000000002e228 < x.im

   1. Initial program 68.1%

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

   3. Simplified 62.6%

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

      1. cube-mult 62.6%

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

      2. associate-*l* 62.6%

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

      3. distribute-lft-out 68.1%

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

      4. pow2 68.1%

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

      5. pow2 68.1%

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

   6. Applied egg-rr 68.1%

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

   8. Applied egg-rr 12.9%

      \[\leadsto x.re \cdot \color{blue}{\left(x.re \cdot -27\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 17.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.9 \cdot 10^{-29}:\\ \;\;\;\;x.re\_m - x.re\_m\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot 29\\ \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (* x.re_s (if (<= x.re_m 2.9e-29) (- x.re_m x.re_m) (* x.re_m 29.0))))

C

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) {
	double tmp;
	if (x_46_re_m <= 2.9e-29) {
		tmp = x_46_re_m - x_46_re_m;
	} else {
		tmp = x_46_re_m * 29.0;
	}
	return x_46_re_s * tmp;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2.9d-29) then
        tmp = x_46re_m - x_46re_m
    else
        tmp = x_46re_m * 29.0d0
    end if
    code = x_46re_s * tmp
end function

Java

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) {
	double tmp;
	if (x_46_re_m <= 2.9e-29) {
		tmp = x_46_re_m - x_46_re_m;
	} else {
		tmp = x_46_re_m * 29.0;
	}
	return x_46_re_s * tmp;
}

Python

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):
	tmp = 0
	if x_46_re_m <= 2.9e-29:
		tmp = x_46_re_m - x_46_re_m
	else:
		tmp = x_46_re_m * 29.0
	return x_46_re_s * tmp

Julia

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)
	tmp = 0.0
	if (x_46_re_m <= 2.9e-29)
		tmp = Float64(x_46_re_m - x_46_re_m);
	else
		tmp = Float64(x_46_re_m * 29.0);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

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)
	tmp = 0.0;
	if (x_46_re_m <= 2.9e-29)
		tmp = x_46_re_m - x_46_re_m;
	else
		tmp = x_46_re_m * 29.0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.9e-29], N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision], N[(x$46$re$95$m * 29.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.90000000000000024e-29

   1. Initial program 84.5%

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

   3. Simplified 84.0%

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

      1. cube-mult 83.9%

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

      2. associate-*l* 83.9%

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

      3. distribute-lft-out 89.2%

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

      4. pow2 89.2%

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

      5. pow2 89.2%

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

   6. Applied egg-rr 89.2%

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

   8. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{x.re - x.re} \]

   if 2.90000000000000024e-29 < x.re

   1. Initial program 74.9%

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

   3. Simplified 69.1%

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

      1. cube-mult 69.0%

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

      2. associate-*l* 69.0%

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

      3. distribute-lft-out 88.1%

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

      4. pow2 88.1%

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

      5. pow2 88.1%

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

   6. Applied egg-rr 88.1%

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

   8. Applied egg-rr 4.7%

      \[\leadsto \color{blue}{x.re + \mathsf{fma}\left(-x.re, -27, x.re\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 4.7%

         \[\leadsto x.re + \color{blue}{\left(\left(-x.re\right) \cdot -27 + x.re\right)} \]

      2. +-commutative 4.7%

         \[\leadsto x.re + \color{blue}{\left(x.re + \left(-x.re\right) \cdot -27\right)} \]

      3. associate-+r+ 4.7%

         \[\leadsto \color{blue}{\left(x.re + x.re\right) + \left(-x.re\right) \cdot -27} \]

      4. count-2 4.7%

         \[\leadsto \color{blue}{2 \cdot x.re} + \left(-x.re\right) \cdot -27 \]

      5. *-commutative 4.7%

         \[\leadsto \color{blue}{x.re \cdot 2} + \left(-x.re\right) \cdot -27 \]

      6. distribute-lft-neg-out 4.7%

         \[\leadsto x.re \cdot 2 + \color{blue}{\left(-x.re \cdot -27\right)} \]

      7. distribute-rgt-neg-in 4.7%

         \[\leadsto x.re \cdot 2 + \color{blue}{x.re \cdot \left(--27\right)} \]

      8. distribute-lft-out 4.7%

         \[\leadsto \color{blue}{x.re \cdot \left(2 + \left(--27\right)\right)} \]

      9. metadata-eval 4.7%

         \[\leadsto x.re \cdot \left(2 + \color{blue}{27}\right) \]

      10. metadata-eval 4.7%

          \[\leadsto x.re \cdot \color{blue}{29} \]

   10. Simplified 4.7%

       \[\leadsto \color{blue}{x.re \cdot 29} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 4.5% accurate, 6.3× speedup

Math

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

FPCore

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) :precision binary64 (* x.re_s (+ x.re_m x.re_m)))

C

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) {
	return x_46_re_s * (x_46_re_m + x_46_re_m);
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * (x_46re_m + x_46re_m)
end function

Java

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) {
	return x_46_re_s * (x_46_re_m + x_46_re_m);
}

Python

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):
	return x_46_re_s * (x_46_re_m + x_46_re_m)

Julia

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)
	return Float64(x_46_re_s * Float64(x_46_re_m + x_46_re_m))
end

MATLAB

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)
	tmp = x_46_re_s * (x_46_re_m + x_46_re_m);
end

Wolfram

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_] := N[(x$46$re$95$s * N[(x$46$re$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 80.0%

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

   1. cube-mult 80.0%

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

   2. associate-*l* 79.9%

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

   3. distribute-lft-out 88.9%

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

   4. pow2 88.9%

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

   5. pow2 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 4.7%

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

Alternative 9: 4.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.re\_m \cdot 29\right) \end{array} \]

FPCore

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) :precision binary64 (* x.re_s (* x.re_m 29.0)))

C

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) {
	return x_46_re_s * (x_46_re_m * 29.0);
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * (x_46re_m * 29.0d0)
end function

Java

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) {
	return x_46_re_s * (x_46_re_m * 29.0);
}

Python

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):
	return x_46_re_s * (x_46_re_m * 29.0)

Julia

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)
	return Float64(x_46_re_s * Float64(x_46_re_m * 29.0))
end

MATLAB

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)
	tmp = x_46_re_s * (x_46_re_m * 29.0);
end

Wolfram

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_] := N[(x$46$re$95$s * N[(x$46$re$95$m * 29.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 80.0%

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

   1. cube-mult 80.0%

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

   2. associate-*l* 79.9%

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

   3. distribute-lft-out 88.9%

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

   4. pow2 88.9%

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

   5. pow2 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 4.7%

   \[\leadsto \color{blue}{x.re + \mathsf{fma}\left(-x.re, -27, x.re\right)} \]
9. Step-by-step derivation

   1. fma-undefine 4.7%

      \[\leadsto x.re + \color{blue}{\left(\left(-x.re\right) \cdot -27 + x.re\right)} \]

   2. +-commutative 4.7%

      \[\leadsto x.re + \color{blue}{\left(x.re + \left(-x.re\right) \cdot -27\right)} \]

   3. associate-+r+ 4.7%

      \[\leadsto \color{blue}{\left(x.re + x.re\right) + \left(-x.re\right) \cdot -27} \]

   4. count-2 4.7%

      \[\leadsto \color{blue}{2 \cdot x.re} + \left(-x.re\right) \cdot -27 \]

   5. *-commutative 4.7%

      \[\leadsto \color{blue}{x.re \cdot 2} + \left(-x.re\right) \cdot -27 \]

   6. distribute-lft-neg-out 4.7%

      \[\leadsto x.re \cdot 2 + \color{blue}{\left(-x.re \cdot -27\right)} \]

   7. distribute-rgt-neg-in 4.7%

      \[\leadsto x.re \cdot 2 + \color{blue}{x.re \cdot \left(--27\right)} \]

   8. distribute-lft-out 4.7%

      \[\leadsto \color{blue}{x.re \cdot \left(2 + \left(--27\right)\right)} \]

   9. metadata-eval 4.7%

      \[\leadsto x.re \cdot \left(2 + \color{blue}{27}\right) \]

   10. metadata-eval 4.7%

       \[\leadsto x.re \cdot \color{blue}{29} \]

10. Simplified 4.7%

    \[\leadsto \color{blue}{x.re \cdot 29} \]
11. Add Preprocessing

Alternative 10: 3.1% accurate, 9.5× speedup

Math

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

FPCore

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) :precision binary64 (* x.re_s (- x.re_m)))

C

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) {
	return x_46_re_s * -x_46_re_m;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * -x_46re_m
end function

Java

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) {
	return x_46_re_s * -x_46_re_m;
}

Python

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):
	return x_46_re_s * -x_46_re_m

Julia

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)
	return Float64(x_46_re_s * Float64(-x_46_re_m))
end

MATLAB

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)
	tmp = x_46_re_s * -x_46_re_m;
end

Wolfram

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_] := N[(x$46$re$95$s * (-x$46$re$95$m)), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 80.0%

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

   1. cube-mult 80.0%

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

   2. associate-*l* 79.9%

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

   3. distribute-lft-out 88.9%

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

   4. pow2 88.9%

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

   5. pow2 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{-27 + \left(-x.re\right)} \]
9. Step-by-step derivation

   1. sub-neg 2.1%

      \[\leadsto \color{blue}{-27 - x.re} \]

10. Simplified 2.1%

    \[\leadsto \color{blue}{-27 - x.re} \]

11. Taylor expanded in x.re around inf 3.2%

    \[\leadsto \color{blue}{-1 \cdot x.re} \]

13. Simplified 3.2%

    \[\leadsto \color{blue}{-x.re} \]
14. Add Preprocessing

Alternative 11: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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) :precision binary64 (* x.re_s 1.0))

C

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) {
	return x_46_re_s * 1.0;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * 1.0d0
end function

Java

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) {
	return x_46_re_s * 1.0;
}

Python

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):
	return x_46_re_s * 1.0

Julia

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)
	return Float64(x_46_re_s * 1.0)
end

MATLAB

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)
	tmp = x_46_re_s * 1.0;
end

Wolfram

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_] := N[(x$46$re$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 80.0%

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

   1. cube-mult 80.0%

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

   2. associate-*l* 79.9%

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

   3. distribute-lft-out 88.9%

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

   4. pow2 88.9%

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

   5. pow2 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{\frac{x.re}{x.re}} \]
9. Step-by-step derivation

   1. *-inverses 2.7%

      \[\leadsto \color{blue}{1} \]

10. Simplified 2.7%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Alternative 12: 2.8% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot -27 \end{array} \]

FPCore

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) :precision binary64 (* x.re_s -27.0))

C

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) {
	return x_46_re_s * -27.0;
}

Fortran

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)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * (-27.0d0)
end function

Java

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) {
	return x_46_re_s * -27.0;
}

Python

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):
	return x_46_re_s * -27.0

Julia

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)
	return Float64(x_46_re_s * -27.0)
end

MATLAB

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)
	tmp = x_46_re_s * -27.0;
end

Wolfram

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_] := N[(x$46$re$95$s * -27.0), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 80.0%

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

   1. cube-mult 80.0%

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

   2. associate-*l* 79.9%

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

   3. distribute-lft-out 88.9%

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

   4. pow2 88.9%

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

   5. pow2 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{-27 + \left(-x.re\right)} \]
9. Step-by-step derivation

   1. sub-neg 2.1%

      \[\leadsto \color{blue}{-27 - x.re} \]

10. Simplified 2.1%

    \[\leadsto \color{blue}{-27 - x.re} \]

11. Taylor expanded in x.re around 0 2.8%

    \[\leadsto \color{blue}{-27} \]
12. Add Preprocessing

Developer Target 1: 88.3% 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 2024165 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

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