math.cube on complex, real part

Percentage Accurate: 82.6% → 99.7%

Time: 8.7s

Alternatives: 10

Speedup: 0.5×

• 45.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 82.6% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.6 \cdot 10^{+79}:\\ \;\;\;\;\left(x.im \cdot \left(x.re\_m \cdot \left(x.re\_m - x.re\_m\right) - x.re\_m \cdot x.im\right) + {x.re\_m}^{3}\right) - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.6e+79)
    (-
     (+
      (* x.im (- (* x.re_m (- x.re_m x.re_m)) (* x.re_m x.im)))
      (pow x.re_m 3.0))
     (* x.im (* (* x.re_m x.im) 2.0)))
    (* (- x.re_m x.im) (* x.re_m (+ 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 <= 2.6e+79) {
		tmp = ((x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_re_m * x_46_im))) + pow(x_46_re_m, 3.0)) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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 <= 2.6d+79) then
        tmp = ((x_46im * ((x_46re_m * (x_46re_m - x_46re_m)) - (x_46re_m * x_46im))) + (x_46re_m ** 3.0d0)) - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else
        tmp = (x_46re_m - x_46im) * (x_46re_m * (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 <= 2.6e+79) {
		tmp = ((x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_re_m * x_46_im))) + Math.pow(x_46_re_m, 3.0)) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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 <= 2.6e+79:
		tmp = ((x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_re_m * x_46_im))) + math.pow(x_46_re_m, 3.0)) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0))
	else:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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 <= 2.6e+79)
		tmp = Float64(Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * Float64(x_46_re_m - x_46_re_m)) - Float64(x_46_re_m * x_46_im))) + (x_46_re_m ^ 3.0)) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * 2.0)));
	else
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * 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 <= 2.6e+79)
		tmp = ((x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_re_m * x_46_im))) + (x_46_re_m ^ 3.0)) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	else
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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, 2.6e+79], N[(N[(N[(x$46$im * N[(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 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.60000000000000015e79

   1. Initial program 86.3%

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

      1. difference-of-squares 87.7%

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

      2. *-commutative 87.7%

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

   4. Applied egg-rr 87.7%

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

   5. Taylor expanded in x.re around 0 87.7%

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

   6. Taylor expanded in x.im around 0 91.9%

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

   if 2.60000000000000015e79 < x.re

   1. Initial program 72.1%

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

      1. difference-of-squares 83.7%

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

      2. *-commutative 83.7%

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

   4. Applied egg-rr 83.7%

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

   5. Taylor expanded in x.re around 0 83.7%

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

      1. cancel-sign-sub-inv 83.7%

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

      2. associate-*l* 83.7%

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

      3. fma-define 83.7%

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

      4. associate-*r* 83.7%

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

      5. distribute-rgt-neg-in 83.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 72.1%

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

      8. sqr-neg 72.1%

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

      9. sqrt-unprod 72.1%

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

      10. add-sqr-sqrt 72.1%

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

      11. associate-*r* 72.1%

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

      12. add-log-exp 72.1%

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

      13. exp-prod 72.1%

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

      14. *-commutative 72.1%

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

      15. exp-lft-sqr 72.1%

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

      16. exp-sum 72.1%

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

      17. *-commutative 72.1%

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

      18. exp-prod 72.1%

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

      19. *-commutative 72.1%

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

   7. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.6 \cdot 10^{+79}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot \left(x.re - x.re\right) - x.re \cdot x.im\right) + {x.re}^{3}\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.6 \cdot 10^{+79}:\\ \;\;\;\;{x.re\_m}^{3} + x.im \cdot \left(x.re\_m \cdot \left(x.re\_m - x.re\_m\right) + x.im \cdot \left(x.re\_m \cdot -2 - x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.6e+79)
    (+
     (pow x.re_m 3.0)
     (*
      x.im
      (+ (* x.re_m (- x.re_m x.re_m)) (* x.im (- (* x.re_m -2.0) x.re_m)))))
    (* (- x.re_m x.im) (* x.re_m (+ 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 <= 2.6e+79) {
		tmp = pow(x_46_re_m, 3.0) + (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) + (x_46_im * ((x_46_re_m * -2.0) - x_46_re_m))));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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 <= 2.6d+79) then
        tmp = (x_46re_m ** 3.0d0) + (x_46im * ((x_46re_m * (x_46re_m - x_46re_m)) + (x_46im * ((x_46re_m * (-2.0d0)) - x_46re_m))))
    else
        tmp = (x_46re_m - x_46im) * (x_46re_m * (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 <= 2.6e+79) {
		tmp = Math.pow(x_46_re_m, 3.0) + (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) + (x_46_im * ((x_46_re_m * -2.0) - x_46_re_m))));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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 <= 2.6e+79:
		tmp = math.pow(x_46_re_m, 3.0) + (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) + (x_46_im * ((x_46_re_m * -2.0) - x_46_re_m))))
	else:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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 <= 2.6e+79)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(x_46_im * Float64(Float64(x_46_re_m * Float64(x_46_re_m - x_46_re_m)) + Float64(x_46_im * Float64(Float64(x_46_re_m * -2.0) - x_46_re_m)))));
	else
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * 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 <= 2.6e+79)
		tmp = (x_46_re_m ^ 3.0) + (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) + (x_46_im * ((x_46_re_m * -2.0) - x_46_re_m))));
	else
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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, 2.6e+79], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(x$46$im * N[(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 * -2.0), $MachinePrecision] - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.60000000000000015e79

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. sqr-neg 86.3%

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

      3. fmm-def 86.2%

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

      4. sqr-neg 86.2%

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

      5. *-commutative 86.2%

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

      6. distribute-rgt-neg-in 86.2%

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

      7. neg-mul-1 86.2%

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

      8. *-commutative 86.2%

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

      9. distribute-lft-in 86.2%

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

      10. distribute-rgt-out 86.2%

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

      11. metadata-eval 86.2%

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

   3. Simplified 86.2%

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

      1. prod-diff 72.6%

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

      2. fmm-def 72.6%

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

      3. difference-of-squares 72.6%

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

      4. fma-define 72.6%

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

      5. pow2 72.6%

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

   6. Applied egg-rr 72.6%

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

   7. Taylor expanded in x.im around 0 91.8%

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

   if 2.60000000000000015e79 < x.re

   1. Initial program 72.1%

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

      1. difference-of-squares 83.7%

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

      2. *-commutative 83.7%

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

   4. Applied egg-rr 83.7%

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

   5. Taylor expanded in x.re around 0 83.7%

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

      1. cancel-sign-sub-inv 83.7%

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

      2. associate-*l* 83.7%

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

      3. fma-define 83.7%

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

      4. associate-*r* 83.7%

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

      5. distribute-rgt-neg-in 83.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 72.1%

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

      8. sqr-neg 72.1%

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

      9. sqrt-unprod 72.1%

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

      10. add-sqr-sqrt 72.1%

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

      11. associate-*r* 72.1%

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

      12. add-log-exp 72.1%

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

      13. exp-prod 72.1%

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

      14. *-commutative 72.1%

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

      15. exp-lft-sqr 72.1%

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

      16. exp-sum 72.1%

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

      17. *-commutative 72.1%

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

      18. exp-prod 72.1%

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

      19. *-commutative 72.1%

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

   7. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.6 \cdot 10^{+79}:\\ \;\;\;\;{x.re}^{3} + x.im \cdot \left(x.re \cdot \left(x.re - x.re\right) + x.im \cdot \left(x.re \cdot -2 - x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.5 \cdot 10^{+79}:\\ \;\;\;\;{x.re\_m}^{3} + x.re\_m \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.5e+79)
    (+ (pow x.re_m 3.0) (* x.re_m (* x.im (* x.im -3.0))))
    (* (- x.re_m x.im) (* x.re_m (+ 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 <= 2.5e+79) {
		tmp = pow(x_46_re_m, 3.0) + (x_46_re_m * (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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 <= 2.5d+79) then
        tmp = (x_46re_m ** 3.0d0) + (x_46re_m * (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = (x_46re_m - x_46im) * (x_46re_m * (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 <= 2.5e+79) {
		tmp = Math.pow(x_46_re_m, 3.0) + (x_46_re_m * (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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 <= 2.5e+79:
		tmp = math.pow(x_46_re_m, 3.0) + (x_46_re_m * (x_46_im * (x_46_im * -3.0)))
	else:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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 <= 2.5e+79)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(x_46_re_m * Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * 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 <= 2.5e+79)
		tmp = (x_46_re_m ^ 3.0) + (x_46_re_m * (x_46_im * (x_46_im * -3.0)));
	else
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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, 2.5e+79], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.5e79

   1. Initial program 86.3%

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

   3. Simplified 83.9%

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

   if 2.5e79 < x.re

   1. Initial program 72.1%

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

      1. difference-of-squares 83.7%

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

      2. *-commutative 83.7%

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

   4. Applied egg-rr 83.7%

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

   5. Taylor expanded in x.re around 0 83.7%

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

      1. cancel-sign-sub-inv 83.7%

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

      2. associate-*l* 83.7%

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

      3. fma-define 83.7%

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

      4. associate-*r* 83.7%

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

      5. distribute-rgt-neg-in 83.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 72.1%

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

      8. sqr-neg 72.1%

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

      9. sqrt-unprod 72.1%

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

      10. add-sqr-sqrt 72.1%

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

      11. associate-*r* 72.1%

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

      12. add-log-exp 72.1%

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

      13. exp-prod 72.1%

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

      14. *-commutative 72.1%

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

      15. exp-lft-sqr 72.1%

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

      16. exp-sum 72.1%

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

      17. *-commutative 72.1%

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

      18. exp-prod 72.1%

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

      19. *-commutative 72.1%

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

   7. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.5 \cdot 10^{+79}:\\ \;\;\;\;{x.re}^{3} + x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 5 \cdot 10^{-30}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) 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))))
       5e-30)
    (-
     (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))
     (* x.im (* (* x.re_m x.im) 2.0)))
    (* (- x.re_m x.im) (* x.re_m (+ 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)))) <= 5e-30) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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)))) <= 5d-30) then
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else
        tmp = (x_46re_m - x_46im) * (x_46re_m * (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)))) <= 5e-30) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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)))) <= 5e-30:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0))
	else:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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)))) <= 5e-30)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * 2.0)));
	else
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * 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)))) <= 5e-30)
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	else
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (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], 5e-30], N[(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] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * 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)) < 4.99999999999999972e-30

   1. Initial program 92.5%

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

      1. difference-of-squares 92.5%

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

      2. *-commutative 92.5%

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

   4. Applied egg-rr 92.5%

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

   5. Taylor expanded in x.re around 0 92.5%

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

   if 4.99999999999999972e-30 < (-.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 68.0%

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

      1. difference-of-squares 76.9%

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

      2. *-commutative 76.9%

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

   4. Applied egg-rr 76.9%

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

   5. Taylor expanded in x.re around 0 76.9%

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

      1. cancel-sign-sub-inv 76.9%

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

      2. associate-*l* 82.1%

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

      3. fma-define 82.1%

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

      4. associate-*r* 82.1%

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

      5. distribute-rgt-neg-in 82.1%

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

      6. add-sqr-sqrt 33.2%

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

      7. sqrt-unprod 80.5%

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

      8. sqr-neg 80.5%

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

      9. sqrt-unprod 51.0%

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

      10. add-sqr-sqrt 61.1%

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

      11. associate-*r* 61.1%

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

      12. add-log-exp 56.9%

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

      13. exp-prod 55.5%

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

      14. *-commutative 55.5%

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

      15. exp-lft-sqr 55.5%

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

      16. exp-sum 55.5%

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

      17. *-commutative 55.5%

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

      18. exp-prod 56.9%

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

      19. *-commutative 56.9%

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

   7. Applied egg-rr 88.0%

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

      1. fma-undefine 88.0%

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

      2. +-rgt-identity 88.0%

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

      3. *-commutative 88.0%

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

   9. Simplified 88.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{-30}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 7.5 \cdot 10^{+191}:\\ \;\;\;\;x.re\_m \cdot \left(-27 + \left(x.re\_m + 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(-54 - x.re\_m\right)\\ \end{array} \end{array} \]

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.im 7.5e+191)
    (* x.re_m (+ -27.0 (+ x.re_m 27.0)))
    (* x.re_m (- -54.0 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_im <= 7.5e+191) {
		tmp = x_46_re_m * (-27.0 + (x_46_re_m + 27.0));
	} else {
		tmp = x_46_re_m * (-54.0 - 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_46im <= 7.5d+191) then
        tmp = x_46re_m * ((-27.0d0) + (x_46re_m + 27.0d0))
    else
        tmp = x_46re_m * ((-54.0d0) - 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_im <= 7.5e+191) {
		tmp = x_46_re_m * (-27.0 + (x_46_re_m + 27.0));
	} else {
		tmp = x_46_re_m * (-54.0 - 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_im <= 7.5e+191:
		tmp = x_46_re_m * (-27.0 + (x_46_re_m + 27.0))
	else:
		tmp = x_46_re_m * (-54.0 - 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_im <= 7.5e+191)
		tmp = Float64(x_46_re_m * Float64(-27.0 + Float64(x_46_re_m + 27.0)));
	else
		tmp = Float64(x_46_re_m * Float64(-54.0 - 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_im <= 7.5e+191)
		tmp = x_46_re_m * (-27.0 + (x_46_re_m + 27.0));
	else
		tmp = x_46_re_m * (-54.0 - 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$46$im, 7.5e+191], N[(x$46$re$95$m * N[(-27.0 + N[(x$46$re$95$m + 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(-54.0 - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.5e191

   1. Initial program 85.9%

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

      1. difference-of-squares 87.6%

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

      2. *-commutative 87.6%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in x.re around 0 45.5%

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

   7. Simplified 16.2%

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

      1. distribute-lft-in 16.2%

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

   9. Applied egg-rr 28.9%

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

       1. +-commutative 28.9%

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

       2. distribute-lft-out 28.9%

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

       3. sub-neg 28.9%

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

       4. metadata-eval 28.9%

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

   11. Simplified 28.9%

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

   if 7.5e191 < x.im

   1. Initial program 66.3%

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

      1. difference-of-squares 81.7%

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

      2. *-commutative 81.7%

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

   4. Applied egg-rr 81.7%

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

   5. Taylor expanded in x.re around 0 81.7%

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

   7. Simplified 13.5%

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

   8. Taylor expanded in x.re around 0 13.5%

      \[\leadsto \color{blue}{x.re \cdot \left(-1 \cdot x.re - 54\right)} \]
   9. Step-by-step derivation

      1. sub-neg 13.5%

         \[\leadsto x.re \cdot \color{blue}{\left(-1 \cdot x.re + \left(-54\right)\right)} \]

      2. metadata-eval 13.5%

         \[\leadsto x.re \cdot \left(-1 \cdot x.re + \color{blue}{-54}\right) \]

      3. +-commutative 13.5%

         \[\leadsto x.re \cdot \color{blue}{\left(-54 + -1 \cdot x.re\right)} \]

      4. mul-1-neg 13.5%

         \[\leadsto x.re \cdot \left(-54 + \color{blue}{\left(-x.re\right)}\right) \]

      5. sub-neg 13.5%

         \[\leadsto x.re \cdot \color{blue}{\left(-54 - x.re\right)} \]

   10. Simplified 13.5%

       \[\leadsto \color{blue}{x.re \cdot \left(-54 - x.re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.5 \cdot 10^{+191}:\\ \;\;\;\;x.re \cdot \left(-27 + \left(x.re + 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(-54 - x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.3 \cdot 10^{-108}:\\ \;\;\;\;x.re\_m \cdot \left(-x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m + -54\right)\\ \end{array} \end{array} \]

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 1.3e-108) (* x.re_m (- x.re_m)) (* x.re_m (+ x.re_m -54.0)))))

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 <= 1.3e-108) {
		tmp = x_46_re_m * -x_46_re_m;
	} else {
		tmp = x_46_re_m * (x_46_re_m + -54.0);
	}
	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 <= 1.3d-108) then
        tmp = x_46re_m * -x_46re_m
    else
        tmp = x_46re_m * (x_46re_m + (-54.0d0))
    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 <= 1.3e-108) {
		tmp = x_46_re_m * -x_46_re_m;
	} else {
		tmp = x_46_re_m * (x_46_re_m + -54.0);
	}
	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 <= 1.3e-108:
		tmp = x_46_re_m * -x_46_re_m
	else:
		tmp = x_46_re_m * (x_46_re_m + -54.0)
	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 <= 1.3e-108)
		tmp = Float64(x_46_re_m * Float64(-x_46_re_m));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m + -54.0));
	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 <= 1.3e-108)
		tmp = x_46_re_m * -x_46_re_m;
	else
		tmp = x_46_re_m * (x_46_re_m + -54.0);
	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, 1.3e-108], N[(x$46$re$95$m * (-x$46$re$95$m)), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m + -54.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.29999999999999992e-108

   1. Initial program 82.8%

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

      1. difference-of-squares 84.5%

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

      2. *-commutative 84.5%

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

   4. Applied egg-rr 84.5%

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

   5. Taylor expanded in x.re around 0 54.4%

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

   7. Simplified 20.2%

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

   8. Taylor expanded in x.re around inf 34.5%

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

      1. mul-1-neg 34.5%

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

   10. Simplified 34.5%

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

   if 1.29999999999999992e-108 < x.re

   1. Initial program 86.0%

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

      1. difference-of-squares 91.8%

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

      2. *-commutative 91.8%

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

   4. Applied egg-rr 91.8%

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

   5. Taylor expanded in x.re around 0 38.9%

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

   7. Simplified 7.7%

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

      1. pow1 7.7%

         \[\leadsto \color{blue}{{\left(x.re \cdot \left(\left(-27 + \left(-x.re\right)\right) + -27\right)\right)}^{1}} \]

      2. +-commutative 7.7%

         \[\leadsto {\left(x.re \cdot \left(\color{blue}{\left(\left(-x.re\right) + -27\right)} + -27\right)\right)}^{1} \]

      3. associate-+l+ 7.7%

         \[\leadsto {\left(x.re \cdot \color{blue}{\left(\left(-x.re\right) + \left(-27 + -27\right)\right)}\right)}^{1} \]

      4. add-sqr-sqrt 0.0%

         \[\leadsto {\left(x.re \cdot \left(\color{blue}{\sqrt{-x.re} \cdot \sqrt{-x.re}} + \left(-27 + -27\right)\right)\right)}^{1} \]

      5. sqrt-unprod 31.7%

         \[\leadsto {\left(x.re \cdot \left(\color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}} + \left(-27 + -27\right)\right)\right)}^{1} \]

      6. sqr-neg 31.7%

         \[\leadsto {\left(x.re \cdot \left(\sqrt{\color{blue}{x.re \cdot x.re}} + \left(-27 + -27\right)\right)\right)}^{1} \]

      7. sqrt-unprod 31.7%

         \[\leadsto {\left(x.re \cdot \left(\color{blue}{\sqrt{x.re} \cdot \sqrt{x.re}} + \left(-27 + -27\right)\right)\right)}^{1} \]

      8. add-sqr-sqrt 31.7%

         \[\leadsto {\left(x.re \cdot \left(\color{blue}{x.re} + \left(-27 + -27\right)\right)\right)}^{1} \]

      9. metadata-eval 31.7%

         \[\leadsto {\left(x.re \cdot \left(x.re + \color{blue}{-54}\right)\right)}^{1} \]

   9. Applied egg-rr 31.7%

      \[\leadsto \color{blue}{{\left(x.re \cdot \left(x.re + -54\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 31.7%

          \[\leadsto \color{blue}{x.re \cdot \left(x.re + -54\right)} \]

   11. Simplified 31.7%

       \[\leadsto \color{blue}{x.re \cdot \left(x.re + -54\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.3 \cdot 10^{-108}:\\ \;\;\;\;x.re \cdot \left(-x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re + -54\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 2.1× speedup

Math

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

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.im) (* x.re_m (+ 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) {
	return x_46_re_s * ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im)));
}

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_46im) * (x_46re_m * (x_46re_m + x_46im)))
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_im) * (x_46_re_m * (x_46_re_m + x_46_im)));
}

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_im) * (x_46_re_m * (x_46_re_m + x_46_im)))

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

Derivation

1. Initial program 83.9%

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

   1. difference-of-squares 87.0%

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

   2. *-commutative 87.0%

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

4. Applied egg-rr 87.0%

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

5. Taylor expanded in x.re around 0 87.0%

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

   1. cancel-sign-sub-inv 87.0%

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

   2. associate-*l* 93.5%

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

   3. fma-define 93.6%

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

   4. associate-*r* 93.6%

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

   5. distribute-rgt-neg-in 93.6%

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

   6. add-sqr-sqrt 44.4%

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

   7. sqrt-unprod 63.0%

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

   8. sqr-neg 63.0%

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

   9. sqrt-unprod 27.5%

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

   10. add-sqr-sqrt 58.5%

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

   11. associate-*r* 58.5%

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

   12. add-log-exp 58.1%

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

   13. exp-prod 58.2%

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

   14. *-commutative 58.2%

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

   15. exp-lft-sqr 58.2%

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

   16. exp-sum 58.2%

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

   17. *-commutative 58.2%

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

   18. exp-prod 58.1%

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

   19. *-commutative 58.1%

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

7. Applied egg-rr 79.6%

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

   1. fma-undefine 79.6%

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

   2. +-rgt-identity 79.6%

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

   3. *-commutative 79.6%

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

9. Simplified 79.6%

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

10. Final simplification 79.6%

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

Alternative 8: 16.0% accurate, 4.8× speedup

Math

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

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 * 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 * -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 83.9%

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

   1. difference-of-squares 87.0%

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

   2. *-commutative 87.0%

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

4. Applied egg-rr 87.0%

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

5. Taylor expanded in x.re around 0 49.2%

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

7. Simplified 15.9%

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

8. Taylor expanded in x.re around inf 25.4%

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

   1. mul-1-neg 25.4%

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

10. Simplified 25.4%

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

11. Final simplification 25.4%

    \[\leadsto x.re \cdot \left(-x.re\right) \]
12. Add Preprocessing

Alternative 9: 3.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.re\_m \cdot -54\right) \end{array} \]

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 -54.0)))

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 * -54.0);
}

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 * (-54.0d0))
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 * -54.0);
}

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 * -54.0)

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 * -54.0))
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 * -54.0);
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 * -54.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.9%

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

   1. difference-of-squares 87.0%

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

   2. *-commutative 87.0%

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

4. Applied egg-rr 87.0%

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

5. Taylor expanded in x.re around 0 49.2%

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

7. Simplified 15.9%

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

8. Taylor expanded in x.re around 0 3.0%

   \[\leadsto \color{blue}{-54 \cdot x.re} \]
9. Step-by-step derivation

   1. *-commutative 3.0%

      \[\leadsto \color{blue}{x.re \cdot -54} \]

10. Simplified 3.0%

    \[\leadsto \color{blue}{x.re \cdot -54} \]

11. Final simplification 3.0%

    \[\leadsto x.re \cdot -54 \]
12. Add Preprocessing

Alternative 10: 2.7% accurate, 19.0× speedup

Math

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

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 83.9%

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

3. Simplified 79.6%

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

   1. +-commutative 79.6%

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

   2. associate-*r* 86.2%

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

   3. fma-define 87.4%

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

6. Applied egg-rr 87.4%

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

7. Taylor expanded in x.re around 0 49.2%

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

9. Simplified 49.2%

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

    1. *-commutative 49.2%

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

    2. unpow2 49.2%

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

    3. associate-*l* 49.1%

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

    4. associate-*l* 55.7%

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

    5. add-cbrt-cube 43.7%

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

    6. add-cbrt-cube 36.0%

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

    7. cbrt-unprod 35.1%

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

    8. pow3 35.1%

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

    9. pow3 35.1%

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

11. Applied egg-rr 35.1%

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

13. Simplified 3.0%

    \[\leadsto \color{blue}{-27} \]

14. Final simplification 3.0%

    \[\leadsto -27 \]
15. Add Preprocessing

Developer target: 87.3% 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 2024130 
(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)))