math.cube on complex, real part

Percentage Accurate: 83.1% → 99.8%

Time: 8.4s

Alternatives: 3

Speedup: 0.6×

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

Specification

Math

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

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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5 \cdot 10^{+100}:\\ \;\;\;\;{x.re\_m}^{3} + \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m - x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 1 x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5e+100)
    (+ (pow x.re_m 3.0) (* (* x.im (* x.re_m x.im)) -3.0))
    (* x.re_m (* (+ x.re_m x.im) (- x.re_m x.im))))))

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 <= 5e+100) {
		tmp = pow(x_46_re_m, 3.0) + ((x_46_im * (x_46_re_m * x_46_im)) * -3.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im));
	}
	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 <= 5d+100) then
        tmp = (x_46re_m ** 3.0d0) + ((x_46im * (x_46re_m * x_46im)) * (-3.0d0))
    else
        tmp = x_46re_m * ((x_46re_m + x_46im) * (x_46re_m - x_46im))
    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 <= 5e+100) {
		tmp = Math.pow(x_46_re_m, 3.0) + ((x_46_im * (x_46_re_m * x_46_im)) * -3.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im));
	}
	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 <= 5e+100:
		tmp = math.pow(x_46_re_m, 3.0) + ((x_46_im * (x_46_re_m * x_46_im)) * -3.0)
	else:
		tmp = x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))
	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 <= 5e+100)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(Float64(x_46_im * Float64(x_46_re_m * x_46_im)) * -3.0));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m - x_46_im)));
	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 <= 5e+100)
		tmp = (x_46_re_m ^ 3.0) + ((x_46_im * (x_46_re_m * x_46_im)) * -3.0);
	else
		tmp = x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im));
	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, 5e+100], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.9999999999999999e100

   1. Initial program 90.0%

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

   3. Simplified 89.1%

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

      1. add-sqr-sqrt 55.2%

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

      2. pow2 55.2%

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

      3. *-commutative 55.2%

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

      4. sqrt-prod 34.1%

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

      5. sqrt-prod 18.5%

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

      6. add-sqr-sqrt 39.1%

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

   6. Applied egg-rr 39.1%

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

      1. *-commutative 39.1%

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

      2. unpow-prod-down 34.1%

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

      3. pow2 34.1%

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

      4. unpow2 34.1%

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

      5. add-sqr-sqrt 89.1%

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

      6. associate-*r* 95.4%

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

   8. Applied egg-rr 95.4%

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

   if 4.9999999999999999e100 < x.re

   1. Initial program 55.8%

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

      1. difference-of-squares 71.2%

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

      2. *-commutative 71.2%

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

   4. Applied egg-rr 71.2%

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

      1. *-un-lft-identity 71.2%

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

      2. *-commutative 71.2%

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

      3. *-un-lft-identity 71.2%

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

      4. distribute-rgt-out 71.2%

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

      5. metadata-eval 71.2%

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

   6. Applied egg-rr 71.2%

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

      1. associate-*l* 71.2%

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

      2. fma-neg 71.2%

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

      3. *-commutative 71.2%

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

      4. *-commutative 71.2%

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

      5. distribute-rgt-neg-in 71.2%

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

      6. distribute-rgt-neg-in 71.2%

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

      7. metadata-eval 71.2%

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

   8. Applied egg-rr 71.2%

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

      1. fma-undefine 71.2%

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

      2. *-commutative 71.2%

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

      3. associate-*l* 71.2%

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

      4. metadata-eval 71.2%

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

      5. distribute-rgt-neg-in 71.2%

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

      6. distribute-lft-neg-in 71.2%

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

      7. mul-1-neg 71.2%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 76.9%

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

      10. mul-1-neg 76.9%

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

      11. mul-1-neg 76.9%

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

      12. sqr-neg 76.9%

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

      13. sqrt-unprod 76.9%

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

      14. add-sqr-sqrt 76.9%

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

      15. metadata-eval 76.9%

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

      16. distribute-rgt-out-- 76.9%

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

      17. add-sqr-sqrt 0.0%

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

      18. *-un-lft-identity 0.0%

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

      19. sqrt-unprod 38.5%

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

      20. mul-1-neg 38.5%

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

      21. mul-1-neg 38.5%

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

      22. sqr-neg 38.5%

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

      23. sqrt-unprod 82.7%

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

      24. add-sqr-sqrt 100.0%

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

      25. unsub-neg 100.0%

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

      26. mul-1-neg 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) + x.im \cdot \left(x.im \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 59.6%

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

       2. unpow2 59.6%

          \[\leadsto \left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) + \color{blue}{{x.im}^{2}} \cdot 0 \]

       3. mul0-rgt 100.0%

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

       4. +-rgt-identity 100.0%

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

       5. *-commutative 100.0%

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

       6. associate-*l* 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5 \cdot 10^{+100}:\\ \;\;\;\;{x.re}^{3} + \left(x.im \cdot \left(x.re \cdot x.im\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m - x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 1 x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       -4e-299)
    (* (* x.im (* x.re_m x.im)) -3.0)
    (* x.re_m (* (+ x.re_m x.im) (- x.re_m x.im))))))

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 * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -4e-299) {
		tmp = (x_46_im * (x_46_re_m * x_46_im)) * -3.0;
	} else {
		tmp = x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im));
	}
	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 * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-4d-299)) then
        tmp = (x_46im * (x_46re_m * x_46im)) * (-3.0d0)
    else
        tmp = x_46re_m * ((x_46re_m + x_46im) * (x_46re_m - x_46im))
    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 * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -4e-299) {
		tmp = (x_46_im * (x_46_re_m * x_46_im)) * -3.0;
	} else {
		tmp = x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im));
	}
	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 * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -4e-299:
		tmp = (x_46_im * (x_46_re_m * x_46_im)) * -3.0
	else:
		tmp = x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))
	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 (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -4e-299)
		tmp = Float64(Float64(x_46_im * Float64(x_46_re_m * x_46_im)) * -3.0);
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m - x_46_im)));
	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 * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -4e-299)
		tmp = (x_46_im * (x_46_re_m * x_46_im)) * -3.0;
	else
		tmp = x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im));
	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[N[(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$46$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], -4e-299], N[(N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $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)) < -3.99999999999999997e-299

   1. Initial program 88.9%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x.re around 0 37.5%

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

      1. add-sqr-sqrt 36.5%

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

      2. pow2 36.5%

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

      3. unpow-prod-down 47.3%

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

   7. Applied egg-rr 47.3%

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

      1. *-commutative 43.3%

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

      2. unpow-prod-down 32.5%

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

      3. pow2 32.5%

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

      4. unpow2 32.5%

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

      5. add-sqr-sqrt 84.7%

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

      6. associate-*r* 95.6%

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

   9. Applied egg-rr 48.3%

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

   if -3.99999999999999997e-299 < (-.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 79.7%

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

      1. difference-of-squares 85.8%

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

      2. *-commutative 85.8%

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

   4. Applied egg-rr 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. *-commutative 85.8%

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

      3. *-un-lft-identity 85.8%

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

      4. distribute-rgt-out 85.8%

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

      5. metadata-eval 85.8%

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

   6. Applied egg-rr 85.8%

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

      1. associate-*l* 87.5%

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

      2. fma-neg 87.5%

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

      3. *-commutative 87.5%

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

      4. *-commutative 87.5%

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

      5. distribute-rgt-neg-in 87.5%

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

      6. distribute-rgt-neg-in 87.5%

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

      7. metadata-eval 87.5%

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

   8. Applied egg-rr 87.5%

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

      1. fma-undefine 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*l* 87.5%

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

      4. metadata-eval 87.5%

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

      5. distribute-rgt-neg-in 87.5%

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

      6. distribute-lft-neg-in 87.5%

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

      7. mul-1-neg 87.5%

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

      8. add-sqr-sqrt 43.0%

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

      9. sqrt-unprod 85.3%

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

      10. mul-1-neg 85.3%

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

      11. mul-1-neg 85.3%

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

      12. sqr-neg 85.3%

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

      13. sqrt-unprod 48.7%

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

      14. add-sqr-sqrt 64.5%

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

      15. metadata-eval 64.5%

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

      16. distribute-rgt-out-- 64.5%

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

      17. add-sqr-sqrt 15.7%

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

      18. *-un-lft-identity 15.7%

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

      19. sqrt-unprod 49.8%

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

      20. mul-1-neg 49.8%

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

      21. mul-1-neg 49.8%

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

      22. sqr-neg 49.8%

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

      23. sqrt-unprod 49.4%

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

      24. add-sqr-sqrt 81.2%

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

      25. unsub-neg 81.2%

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

      26. mul-1-neg 81.2%

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

   10. Applied egg-rr 81.2%

       \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) + x.im \cdot \left(x.im \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 61.1%

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

       2. unpow2 61.1%

          \[\leadsto \left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) + \color{blue}{{x.im}^{2}} \cdot 0 \]

       3. mul0-rgt 81.2%

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

       4. +-rgt-identity 81.2%

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

       5. *-commutative 81.2%

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

       6. associate-*l* 81.0%

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

   12. Simplified 81.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot x.im\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.9% accurate, 2.7× speedup

Math

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

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 1 x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (* x.re_s (* (* x.im (* x.re_m x.im)) -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) {
	return x_46_re_s * ((x_46_im * (x_46_re_m * x_46_im)) * -3.0);
}

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_46im * (x_46re_m * x_46im)) * (-3.0d0))
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_im * (x_46_re_m * x_46_im)) * -3.0);
}

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_im * (x_46_re_m * x_46_im)) * -3.0)

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(Float64(x_46_im * Float64(x_46_re_m * x_46_im)) * -3.0))
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_im * (x_46_re_m * x_46_im)) * -3.0);
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[(N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

3. Simplified 80.7%

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

5. Taylor expanded in x.re around 0 50.0%

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

   1. add-sqr-sqrt 23.9%

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

   2. pow2 23.9%

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

   3. unpow-prod-down 27.9%

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

7. Applied egg-rr 27.9%

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

   1. *-commutative 40.9%

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

   2. unpow-prod-down 36.9%

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

   3. pow2 36.9%

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

   4. unpow2 36.9%

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

   5. add-sqr-sqrt 80.8%

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

   6. associate-*r* 85.8%

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

9. Applied egg-rr 55.0%

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

10. Final simplification 55.0%

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

Developer target: 87.4% 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 2024052 
(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)))