math.cube on complex, imaginary part

Percentage Accurate: 82.0% → 99.8%

Time: 10.5s

Alternatives: 9

Speedup: 1.5×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{+104} \lor \neg \left(x.im \leq 2 \cdot 10^{+98}\right):\\ \;\;\;\;x.im \cdot \left(2 + \left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -4e+104) (not (<= x.im 2e+98)))
   (* x.im (+ 2.0 (* (+ x.re x.im) (- x.re x.im))))
   (- (* x.re (* x.re (* x.im 3.0))) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -4e+104) || !(x_46_im <= 2e+98)) {
		tmp = x_46_im * (2.0 + ((x_46_re + x_46_im) * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - pow(x_46_im, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-4d+104)) .or. (.not. (x_46im <= 2d+98))) then
        tmp = x_46im * (2.0d0 + ((x_46re + x_46im) * (x_46re - x_46im)))
    else
        tmp = (x_46re * (x_46re * (x_46im * 3.0d0))) - (x_46im ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -4e+104) || !(x_46_im <= 2e+98)) {
		tmp = x_46_im * (2.0 + ((x_46_re + x_46_im) * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -4e+104) or not (x_46_im <= 2e+98):
		tmp = x_46_im * (2.0 + ((x_46_re + x_46_im) * (x_46_re - x_46_im)))
	else:
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -4e+104) || !(x_46_im <= 2e+98))
		tmp = Float64(x_46_im * Float64(2.0 + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im))));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0))) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -4e+104) || ~((x_46_im <= 2e+98)))
		tmp = x_46_im * (2.0 + ((x_46_re + x_46_im) * (x_46_re - x_46_im)));
	else
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -4e+104], N[Not[LessEqual[x$46$im, 2e+98]], $MachinePrecision]], N[(x$46$im * N[(2.0 + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4e104 or 2e98 < x.im

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. *-commutative 80.5%

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

      3. difference-of-squares 87.8%

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

      4. associate-*l* 87.8%

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

      5. fma-def 87.8%

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

      6. *-commutative 87.8%

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

      7. *-commutative 87.8%

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

      8. *-commutative 87.8%

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

      9. distribute-lft-out 87.8%

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

   3. Simplified 87.8%

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

      1. fma-udef 87.8%

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

      2. *-commutative 87.8%

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

      3. associate-*r* 87.8%

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

      4. difference-of-squares 80.5%

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

      5. distribute-lft-in 80.5%

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

      6. *-commutative 80.5%

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

      7. *-commutative 80.5%

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

      8. +-commutative 80.5%

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

   5. Applied egg-rr 86.6%

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

      1. difference-of-squares 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. associate-+l+ 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

       2. associate-+r+ 100.0%

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

       3. count-2 100.0%

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

       4. *-commutative 100.0%

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

       5. distribute-lft-out 100.0%

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

       6. +-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -4e104 < x.im < 2e98

   1. Initial program 94.4%

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

      1. +-commutative 94.4%

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

      2. *-commutative 94.4%

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

      3. distribute-lft-out 94.4%

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

      4. associate-*l* 94.3%

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

      5. *-commutative 94.3%

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

      6. distribute-lft-out 94.4%

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

      7. associate-+r- 94.4%

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

      8. distribute-lft-out-- 94.4%

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

   3. Simplified 94.5%

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

      1. sub-neg 94.5%

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

      2. associate-*l* 94.6%

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

      3. associate-*l* 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unsub-neg 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{+104} \lor \neg \left(x.im \leq 2 \cdot 10^{+98}\right):\\ \;\;\;\;x.im \cdot \left(2 + \left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(2 + \left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (+
       (* x.im (- (* x.re x.re) (* x.im x.im)))
       (* x.re (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (fma (+ x.re x.im) (* x.im (- x.re x.im)) (* x.re (* x.re (+ x.im x.im))))
   (* x.im (+ 2.0 (* (+ x.re x.im) (- x.re x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = fma((x_46_re + x_46_im), (x_46_im * (x_46_re - x_46_im)), (x_46_re * (x_46_re * (x_46_im + x_46_im))));
	} else {
		tmp = x_46_im * (2.0 + ((x_46_re + x_46_im) * (x_46_re - x_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = fma(Float64(x_46_re + x_46_im), Float64(x_46_im * Float64(x_46_re - x_46_im)), Float64(x_46_re * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_im * Float64(2.0 + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(2.0 + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - 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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 95.9%

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

      1. *-commutative 95.9%

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

      2. *-commutative 95.9%

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

      3. difference-of-squares 95.9%

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

      4. associate-*l* 99.7%

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

      5. fma-def 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. difference-of-squares 37.5%

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

      4. associate-*l* 37.5%

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

      5. fma-def 37.5%

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

      6. *-commutative 37.5%

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

      7. *-commutative 37.5%

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

      8. *-commutative 37.5%

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

      9. distribute-lft-out 37.5%

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

   3. Simplified 37.5%

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

      1. fma-udef 37.5%

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

      2. *-commutative 37.5%

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

      3. associate-*r* 37.5%

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

      4. difference-of-squares 0.0%

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

      5. distribute-lft-in 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. +-commutative 0.0%

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

   5. Applied egg-rr 31.3%

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

      1. difference-of-squares 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. associate-+l+ 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

       2. associate-+r+ 100.0%

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

       3. count-2 100.0%

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

       4. *-commutative 100.0%

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

       5. distribute-lft-out 100.0%

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

       6. +-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(2 + \left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 3: 91.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.1 \lor \neg \left(x.im \leq 2200000\right):\\ \;\;\;\;x.im \cdot \left(2 + \left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -2.1) (not (<= x.im 2200000.0)))
   (* x.im (+ 2.0 (* (+ x.re x.im) (- x.re x.im))))
   (* x.re (* (* x.re x.im) 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.1) || !(x_46_im <= 2200000.0)) {
		tmp = x_46_im * (2.0 + ((x_46_re + x_46_im) * (x_46_re - x_46_im)));
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-2.1d0)) .or. (.not. (x_46im <= 2200000.0d0))) then
        tmp = x_46im * (2.0d0 + ((x_46re + x_46im) * (x_46re - x_46im)))
    else
        tmp = x_46re * ((x_46re * x_46im) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.1) || !(x_46_im <= 2200000.0)) {
		tmp = x_46_im * (2.0 + ((x_46_re + x_46_im) * (x_46_re - x_46_im)));
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -2.1) or not (x_46_im <= 2200000.0):
		tmp = x_46_im * (2.0 + ((x_46_re + x_46_im) * (x_46_re - x_46_im)))
	else:
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -2.1) || !(x_46_im <= 2200000.0))
		tmp = Float64(x_46_im * Float64(2.0 + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im))));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -2.1) || ~((x_46_im <= 2200000.0)))
		tmp = x_46_im * (2.0 + ((x_46_re + x_46_im) * (x_46_re - x_46_im)));
	else
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -2.1], N[Not[LessEqual[x$46$im, 2200000.0]], $MachinePrecision]], N[(x$46$im * N[(2.0 + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.10000000000000009 or 2.2e6 < x.im

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. *-commutative 86.6%

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

      3. difference-of-squares 91.6%

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

      4. associate-*l* 91.6%

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

      5. fma-def 91.6%

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

      6. *-commutative 91.6%

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

      7. *-commutative 91.6%

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

      8. *-commutative 91.6%

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

      9. distribute-lft-out 91.6%

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

   3. Simplified 91.6%

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

      1. fma-udef 91.6%

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

      2. *-commutative 91.6%

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

      3. associate-*r* 91.6%

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

      4. difference-of-squares 86.6%

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

      5. distribute-lft-in 86.6%

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

      6. *-commutative 86.6%

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

      7. *-commutative 86.6%

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

      8. +-commutative 86.6%

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

   5. Applied egg-rr 88.6%

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

      1. difference-of-squares 97.8%

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

   7. Applied egg-rr 97.8%

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

      1. *-un-lft-identity 97.8%

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

      2. associate-+l+ 97.8%

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

   9. Applied egg-rr 97.8%

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

       1. *-lft-identity 97.8%

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

       2. associate-+r+ 97.8%

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

       3. count-2 97.8%

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

       4. *-commutative 97.8%

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

       5. distribute-lft-out 97.8%

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

       6. +-commutative 97.8%

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

   11. Simplified 97.8%

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

   if -2.10000000000000009 < x.im < 2.2e6

   1. Initial program 92.9%

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

      1. difference-of-squares 20.5%

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

   3. Applied egg-rr 92.9%

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

   4. Taylor expanded in x.re around 0 92.9%

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

   5. Taylor expanded in x.re around inf 76.1%

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

      1. distribute-rgt-in 76.0%

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

      2. associate-*r* 76.0%

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

      3. distribute-rgt1-in 76.0%

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

      4. metadata-eval 76.0%

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

      5. unpow2 76.0%

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

   7. Simplified 76.0%

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

      1. associate-*r* 76.1%

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

      2. associate-*r* 82.8%

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

      3. *-commutative 82.8%

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

      4. associate-*r* 82.7%

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

      5. *-commutative 82.7%

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

      6. expm1-log1p-u 66.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right)\right)} \]

      7. expm1-udef 45.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right)} - 1} \]

      8. *-commutative 45.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.im \cdot \left(3 \cdot x.re\right)\right) \cdot x.re}\right)} - 1 \]

      9. associate-*l* 41.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(3 \cdot x.re\right) \cdot x.re\right)}\right)} - 1 \]

      10. *-commutative 41.2%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{\left(x.re \cdot 3\right)} \cdot x.re\right)\right)} - 1 \]

   9. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(\left(x.re \cdot 3\right) \cdot x.re\right)\right)} - 1} \]

   11. Simplified 82.7%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.1 \lor \neg \left(x.im \leq 2200000\right):\\ \;\;\;\;x.im \cdot \left(2 + \left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \]

Alternative 4: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 2.2 \cdot 10^{+152}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 2.2e+152)
   (* x.im (- (* (* x.re x.re) 3.0) (* x.im x.im)))
   (* x.re (* x.im (* x.re 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 2.2e+152) {
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 2.2d+152) then
        tmp = x_46im * (((x_46re * x_46re) * 3.0d0) - (x_46im * x_46im))
    else
        tmp = x_46re * (x_46im * (x_46re * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 2.2e+152) {
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 2.2e+152:
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im))
	else:
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 2.2e+152)
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) * 3.0) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 2.2e+152)
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	else
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 2.2e+152], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.1999999999999998e152

   1. Initial program 94.1%

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

      1. difference-of-squares 53.8%

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

   3. Applied egg-rr 95.0%

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

   4. Taylor expanded in x.re around 0 95.0%

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

   5. Taylor expanded in x.im around 0 94.9%

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

      1. *-commutative 94.9%

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

      2. associate-*r* 94.9%

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

      3. unpow2 94.9%

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

      4. associate-*l* 94.9%

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

   7. Simplified 94.9%

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

      1. +-commutative 94.9%

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

      2. *-commutative 94.9%

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

      3. distribute-lft-out 97.2%

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

      4. difference-of-squares 96.3%

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

   9. Applied egg-rr 96.3%

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

   11. Simplified 96.3%

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

   if 2.1999999999999998e152 < x.re

   1. Initial program 58.7%

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

      1. difference-of-squares 78.7%

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

   3. Applied egg-rr 72.0%

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

   4. Taylor expanded in x.re around 0 72.0%

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

   5. Taylor expanded in x.re around inf 72.0%

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

      1. unpow2 72.0%

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

      2. distribute-rgt1-in 72.0%

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

      3. metadata-eval 72.0%

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

      4. associate-*r* 93.2%

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

      5. *-commutative 93.2%

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

      6. associate-*l* 93.2%

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

      7. *-commutative 93.2%

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

      8. associate-*l* 93.3%

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

      9. *-commutative 93.3%

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

   7. Simplified 93.3%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.2 \cdot 10^{+152}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 5: 50.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. difference-of-squares 56.7%

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

3. Applied egg-rr 92.3%

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

4. Taylor expanded in x.re around 0 92.3%

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

5. Taylor expanded in x.re around inf 51.7%

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

   1. distribute-rgt-in 51.6%

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

   2. associate-*r* 51.6%

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

   3. distribute-rgt1-in 51.6%

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

   4. metadata-eval 51.6%

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

   5. unpow2 51.6%

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

7. Simplified 51.6%

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

8. Final simplification 51.6%

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

Alternative 6: 50.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * ((x_46_re * x_46_re) * 3.0);
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = x_46im * ((x_46re * x_46re) * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. difference-of-squares 56.7%

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

3. Applied egg-rr 92.3%

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

4. Taylor expanded in x.re around 0 92.3%

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

5. Taylor expanded in x.re around inf 51.7%

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

   1. distribute-rgt1-in 51.7%

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

   2. metadata-eval 51.7%

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

   3. *-commutative 51.7%

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

   4. *-commutative 51.7%

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

   5. associate-*r* 51.7%

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

   6. unpow2 51.7%

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

7. Simplified 51.7%

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

8. Final simplification 51.7%

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

Alternative 7: 56.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_im * (x_46_re * 3.0));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = x_46re * (x_46im * (x_46re * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. difference-of-squares 56.7%

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

3. Applied egg-rr 92.3%

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

4. Taylor expanded in x.re around 0 92.3%

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

5. Taylor expanded in x.re around inf 51.7%

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

   1. unpow2 51.7%

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

   2. distribute-rgt1-in 51.7%

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

   3. metadata-eval 51.7%

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

   4. associate-*r* 55.2%

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

   5. *-commutative 55.2%

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

   6. associate-*l* 55.2%

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

   7. *-commutative 55.2%

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

   8. associate-*l* 55.2%

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

   9. *-commutative 55.2%

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

7. Simplified 55.2%

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

8. Final simplification 55.2%

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

Alternative 8: 56.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return x_46_re * ((x_46_re * x_46_im) * 3.0);
}

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) * 3.0d0)
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) * 3.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. difference-of-squares 56.7%

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

3. Applied egg-rr 92.3%

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

4. Taylor expanded in x.re around 0 92.3%

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

5. Taylor expanded in x.re around inf 51.7%

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

   1. distribute-rgt-in 51.6%

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

   2. associate-*r* 51.6%

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

   3. distribute-rgt1-in 51.6%

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

   4. metadata-eval 51.6%

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

   5. unpow2 51.6%

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

7. Simplified 51.6%

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

   1. associate-*r* 51.7%

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

   2. associate-*r* 55.2%

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

   3. *-commutative 55.2%

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

   4. associate-*r* 55.2%

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

   5. *-commutative 55.2%

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

   6. expm1-log1p-u 40.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right)\right)} \]

   7. expm1-udef 29.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right)} - 1} \]

   8. *-commutative 29.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.im \cdot \left(3 \cdot x.re\right)\right) \cdot x.re}\right)} - 1 \]

   9. associate-*l* 27.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(3 \cdot x.re\right) \cdot x.re\right)}\right)} - 1 \]

   10. *-commutative 27.3%

       \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{\left(x.re \cdot 3\right)} \cdot x.re\right)\right)} - 1 \]

9. Applied egg-rr 27.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(\left(x.re \cdot 3\right) \cdot x.re\right)\right)} - 1} \]

11. Simplified 55.2%

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

12. Final simplification 55.2%

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

Alternative 9: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return -3.0;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -3.0d0
end function

Java

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

Python

def code(x_46_re, x_46_im):
	return -3.0

Julia

function code(x_46_re, x_46_im)
	return -3.0
end

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := -3.0

Derivation

1. Initial program 89.9%

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

   1. *-commutative 89.9%

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

   2. *-commutative 89.9%

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

   3. difference-of-squares 92.3%

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

   4. associate-*l* 95.8%

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

   5. fma-def 95.9%

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

   6. *-commutative 95.9%

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

   7. *-commutative 95.9%

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

   8. *-commutative 95.9%

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

   9. distribute-lft-out 95.9%

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

3. Simplified 95.9%

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

4. Taylor expanded in x.re around 0 57.8%

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]

6. Simplified 2.7%

   \[\leadsto \color{blue}{-3} \]

7. Final simplification 2.7%

   \[\leadsto -3 \]

Developer target: 91.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023275 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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