math.cube on complex, real part

Percentage Accurate: 83.3% → 98.3%

Time: 9.8s

Alternatives: 11

Speedup: 2.1×

• 44.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.3% 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: 98.3% accurate, 0.1× speedup

Math

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

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* x.re (- (* x.re x.re) (* x.im x.im)))
       (* x.im (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (fma
    (- x.re x.im)
    (* x.re (+ x.re x.im))
    (* x.im (* x.re (- (- x.im) x.im))))
   (* (* x.re x.re) (- x.re x.im))))

C

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

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * 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_re * Float64(x_46_re + x_46_im)), Float64(x_46_im * Float64(x_46_re * Float64(Float64(-x_46_im) - x_46_im))));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im));
	end
	return tmp
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * 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$46$re * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(x$46$re * N[((-x$46$im) - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $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)) < +inf.0

   1. Initial program 94.4%

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

      1. sqr-neg 94.4%

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

      2. difference-of-squares 94.4%

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

      3. sub-neg 94.4%

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

      4. associate-*l* 99.8%

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

      5. sub-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-out 99.8%

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

   3. Simplified 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. fma-def 99.8%

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

      3. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. sqr-neg 0.0%

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

      2. difference-of-squares 38.7%

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

      3. sub-neg 38.7%

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

      4. associate-*l* 38.7%

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

      5. sub-neg 38.7%

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

      6. remove-double-neg 38.7%

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

      7. +-commutative 38.7%

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

      8. *-commutative 38.7%

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

      9. *-commutative 38.7%

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

      10. distribute-rgt-out 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in x.re around inf 19.4%

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

      1. unpow2 19.4%

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

   6. Simplified 19.4%

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

   7. Taylor expanded in x.re around inf 29.0%

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

      1. unpow3 29.0%

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

      2. mul-1-neg 29.0%

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

      3. unpow2 29.0%

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

      4. distribute-rgt-neg-in 29.0%

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

      5. distribute-lft-in 80.6%

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

      6. sub-neg 80.6%

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

      7. *-commutative 80.6%

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

   9. Simplified 80.6%

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

4. Final simplification 97.5%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* x.re (- (* x.re x.re) (* x.im x.im)))
       (* x.im (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (- (* (- x.re x.im) (* x.re (+ x.re x.im))) (* x.im (* x.re (+ x.im x.im))))
   (* (* x.re x.re) (- x.re x.im))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if ((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= math.inf:
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	else:
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = Float64(Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re * Float64(x_46_re + x_46_im))) - Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Inf)
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	else
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $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)) < +inf.0

   1. Initial program 94.4%

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

      1. sqr-neg 94.4%

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

      2. difference-of-squares 94.4%

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

      3. sub-neg 94.4%

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

      4. associate-*l* 99.8%

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

      5. sub-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-out 99.8%

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

   3. Simplified 99.8%

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

   if +inf.0 < (-.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 0.0%

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

      1. sqr-neg 0.0%

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

      2. difference-of-squares 38.7%

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

      3. sub-neg 38.7%

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

      4. associate-*l* 38.7%

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

      5. sub-neg 38.7%

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

      6. remove-double-neg 38.7%

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

      7. +-commutative 38.7%

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

      8. *-commutative 38.7%

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

      9. *-commutative 38.7%

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

      10. distribute-rgt-out 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in x.re around inf 19.4%

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

      1. unpow2 19.4%

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

   6. Simplified 19.4%

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

   7. Taylor expanded in x.re around inf 29.0%

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

      1. unpow3 29.0%

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

      2. mul-1-neg 29.0%

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

      3. unpow2 29.0%

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

      4. distribute-rgt-neg-in 29.0%

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

      5. distribute-lft-in 80.6%

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

      6. sub-neg 80.6%

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

      7. *-commutative 80.6%

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

   9. Simplified 80.6%

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

4. Final simplification 97.5%

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

Alternative 3: 96.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(\left(-x.im\right) - x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 7.8e+153)
   (* x.re (+ (* x.re x.re) (* (* x.im x.im) -3.0)))
   (- (* x.im (* x.re (- (- x.im) x.im))) (* x.im (* x.re x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 7.8e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + ((x_46_im * x_46_im) * -3.0));
	} else {
		tmp = (x_46_im * (x_46_re * (-x_46_im - x_46_im))) - (x_46_im * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= 7.8d+153) then
        tmp = x_46re * ((x_46re * x_46re) + ((x_46im * x_46im) * (-3.0d0)))
    else
        tmp = (x_46im * (x_46re * (-x_46im - x_46im))) - (x_46im * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 7.8e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + ((x_46_im * x_46_im) * -3.0));
	} else {
		tmp = (x_46_im * (x_46_re * (-x_46_im - x_46_im))) - (x_46_im * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 7.8e+153:
		tmp = x_46_re * ((x_46_re * x_46_re) + ((x_46_im * x_46_im) * -3.0))
	else:
		tmp = (x_46_im * (x_46_re * (-x_46_im - x_46_im))) - (x_46_im * (x_46_re * x_46_im))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 7.8e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(Float64(x_46_im * x_46_im) * -3.0)));
	else
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * Float64(Float64(-x_46_im) - x_46_im))) - Float64(x_46_im * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 7.8e+153)
		tmp = x_46_re * ((x_46_re * x_46_re) + ((x_46_im * x_46_im) * -3.0));
	else
		tmp = (x_46_im * (x_46_re * (-x_46_im - x_46_im))) - (x_46_im * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 7.8e+153], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(x$46$re * N[((-x$46$im) - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.79999999999999966e153

   1. Initial program 88.0%

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

      1. sqr-neg 88.0%

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

      2. difference-of-squares 91.1%

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

      3. sub-neg 91.1%

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

      4. associate-*l* 94.4%

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

      5. sub-neg 94.4%

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

      6. remove-double-neg 94.4%

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

      7. +-commutative 94.4%

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

      8. *-commutative 94.4%

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

      9. *-commutative 94.4%

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

      10. distribute-rgt-out 94.4%

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

   3. Simplified 94.4%

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

      1. cancel-sign-sub-inv 94.4%

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

      2. fma-def 94.4%

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

      3. *-commutative 94.4%

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

   5. Applied egg-rr 94.4%

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

      1. fma-udef 94.4%

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

   7. Applied egg-rr 94.4%

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

      1. cancel-sign-sub-inv 94.4%

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

      2. associate-*r* 94.4%

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

      3. *-commutative 94.4%

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

      4. associate-*r* 91.1%

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

      5. count-2 91.1%

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

      6. associate-*r* 91.1%

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

      7. cancel-sign-sub-inv 91.1%

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

      8. *-commutative 91.1%

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

      9. associate-*r* 91.1%

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

      10. distribute-rgt-out 95.1%

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

      11. +-commutative 95.1%

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

      12. distribute-lft-neg-in 95.1%

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

      13. metadata-eval 95.1%

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

   9. Simplified 95.1%

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

       1. distribute-lft-in 91.1%

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

       2. *-commutative 91.1%

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

       3. +-commutative 91.1%

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

       4. difference-of-squares 88.0%

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

       5. associate-*r* 88.0%

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

   11. Applied egg-rr 88.0%

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

       1. associate-*l* 88.0%

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

       2. distribute-lft-in 92.0%

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

       3. unpow2 92.0%

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

       4. sub-neg 92.0%

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

       5. mul-1-neg 92.0%

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

       6. unpow2 92.0%

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

       7. associate-+r+ 92.0%

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

       8. unpow2 92.0%

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

       9. distribute-rgt-out 92.0%

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

       10. metadata-eval 92.0%

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

       11. *-commutative 92.0%

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

       12. unpow2 92.0%

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

       13. unpow2 92.0%

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

       14. unpow2 92.0%

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

   13. Simplified 92.0%

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

   if 7.79999999999999966e153 < x.im

   1. Initial program 49.3%

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

      1. sqr-neg 49.3%

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

      2. difference-of-squares 64.4%

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

      3. sub-neg 64.4%

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

      4. associate-*l* 78.7%

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

      5. sub-neg 78.7%

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

      6. remove-double-neg 78.7%

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

      7. +-commutative 78.7%

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

      8. *-commutative 78.7%

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

      9. *-commutative 78.7%

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

      10. distribute-rgt-out 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in x.re around 0 64.4%

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

      1. associate-*r* 64.4%

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

      2. mul-1-neg 64.4%

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

      3. unpow2 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in x.re around 0 64.4%

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

      1. unpow2 64.4%

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

      2. associate-*r* 64.4%

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

      3. neg-mul-1 64.4%

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

      4. associate-*r* 78.7%

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

      5. *-commutative 78.7%

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

      6. distribute-lft-neg-out 78.7%

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

      7. *-commutative 78.7%

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

   9. Simplified 78.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(\left(-x.im\right) - x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 4: 96.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 7.8e+153)
   (* x.re (+ (* x.re x.re) (* (* x.im x.im) -3.0)))
   (* (* x.re x.im) (* x.im -3.0))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 7.8e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + ((x_46_im * x_46_im) * -3.0));
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= 7.8d+153) then
        tmp = x_46re * ((x_46re * x_46re) + ((x_46im * x_46im) * (-3.0d0)))
    else
        tmp = (x_46re * x_46im) * (x_46im * (-3.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 7.8e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + ((x_46_im * x_46_im) * -3.0));
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 7.8e+153:
		tmp = x_46_re * ((x_46_re * x_46_re) + ((x_46_im * x_46_im) * -3.0))
	else:
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 7.8e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(Float64(x_46_im * x_46_im) * -3.0)));
	else
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_im * -3.0));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 7.8e+153)
		tmp = x_46_re * ((x_46_re * x_46_re) + ((x_46_im * x_46_im) * -3.0));
	else
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 7.8e+153], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.79999999999999966e153

   1. Initial program 88.0%

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

      1. sqr-neg 88.0%

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

      2. difference-of-squares 91.1%

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

      3. sub-neg 91.1%

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

      4. associate-*l* 94.4%

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

      5. sub-neg 94.4%

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

      6. remove-double-neg 94.4%

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

      7. +-commutative 94.4%

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

      8. *-commutative 94.4%

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

      9. *-commutative 94.4%

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

      10. distribute-rgt-out 94.4%

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

   3. Simplified 94.4%

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

      1. cancel-sign-sub-inv 94.4%

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

      2. fma-def 94.4%

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

      3. *-commutative 94.4%

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

   5. Applied egg-rr 94.4%

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

      1. fma-udef 94.4%

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

   7. Applied egg-rr 94.4%

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

      1. cancel-sign-sub-inv 94.4%

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

      2. associate-*r* 94.4%

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

      3. *-commutative 94.4%

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

      4. associate-*r* 91.1%

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

      5. count-2 91.1%

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

      6. associate-*r* 91.1%

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

      7. cancel-sign-sub-inv 91.1%

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

      8. *-commutative 91.1%

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

      9. associate-*r* 91.1%

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

      10. distribute-rgt-out 95.1%

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

      11. +-commutative 95.1%

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

      12. distribute-lft-neg-in 95.1%

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

      13. metadata-eval 95.1%

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

   9. Simplified 95.1%

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

       1. distribute-lft-in 91.1%

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

       2. *-commutative 91.1%

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

       3. +-commutative 91.1%

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

       4. difference-of-squares 88.0%

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

       5. associate-*r* 88.0%

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

   11. Applied egg-rr 88.0%

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

       1. associate-*l* 88.0%

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

       2. distribute-lft-in 92.0%

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

       3. unpow2 92.0%

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

       4. sub-neg 92.0%

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

       5. mul-1-neg 92.0%

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

       6. unpow2 92.0%

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

       7. associate-+r+ 92.0%

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

       8. unpow2 92.0%

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

       9. distribute-rgt-out 92.0%

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

       10. metadata-eval 92.0%

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

       11. *-commutative 92.0%

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

       12. unpow2 92.0%

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

       13. unpow2 92.0%

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

       14. unpow2 92.0%

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

   13. Simplified 92.0%

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

   if 7.79999999999999966e153 < x.im

   1. Initial program 49.3%

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

      1. sqr-neg 49.3%

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

      2. difference-of-squares 64.4%

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

      3. sub-neg 64.4%

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

      4. associate-*l* 78.7%

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

      5. sub-neg 78.7%

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

      6. remove-double-neg 78.7%

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

      7. +-commutative 78.7%

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

      8. *-commutative 78.7%

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

      9. *-commutative 78.7%

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

      10. distribute-rgt-out 78.7%

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

   3. Simplified 78.7%

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

      1. cancel-sign-sub-inv 78.7%

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

      2. fma-def 78.7%

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

      3. *-commutative 78.7%

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

   5. Applied egg-rr 78.7%

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

   6. Taylor expanded in x.re around 0 64.4%

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

      1. unpow2 64.4%

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

      2. unpow2 64.4%

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

      3. distribute-rgt-out 64.4%

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

      4. metadata-eval 64.4%

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

      5. associate-*r* 64.4%

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

      6. *-commutative 64.4%

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

      7. associate-*l* 78.7%

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

      8. *-commutative 78.7%

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

   8. Simplified 78.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \]

Alternative 5: 83.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -7 \cdot 10^{-86}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -7e-86)
   (* x.re (* x.re x.re))
   (if (<= x.re 4.8e+14)
     (* (* x.re x.im) (* x.im -3.0))
     (* (* x.re x.re) (- x.re x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -7e-86) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else if (x_46_re <= 4.8e+14) {
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= (-7d-86)) then
        tmp = x_46re * (x_46re * x_46re)
    else if (x_46re <= 4.8d+14) then
        tmp = (x_46re * x_46im) * (x_46im * (-3.0d0))
    else
        tmp = (x_46re * x_46re) * (x_46re - x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -7e-86) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else if (x_46_re <= 4.8e+14) {
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -7e-86:
		tmp = x_46_re * (x_46_re * x_46_re)
	elif x_46_re <= 4.8e+14:
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0)
	else:
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -7e-86)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	elseif (x_46_re <= 4.8e+14)
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_im * -3.0));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -7e-86)
		tmp = x_46_re * (x_46_re * x_46_re);
	elseif (x_46_re <= 4.8e+14)
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	else
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -7e-86], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4.8e+14], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -7.00000000000000041e-86

   1. Initial program 76.5%

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

      1. sqr-neg 76.5%

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

      2. difference-of-squares 82.6%

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

      3. sub-neg 82.6%

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

      4. associate-*l* 82.6%

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

      5. sub-neg 82.6%

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

      6. remove-double-neg 82.6%

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

      7. +-commutative 82.6%

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

      8. *-commutative 82.6%

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

      9. *-commutative 82.6%

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

      10. distribute-rgt-out 82.6%

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

   3. Simplified 82.6%

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

      1. cancel-sign-sub-inv 82.6%

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

      2. fma-def 82.6%

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

      3. *-commutative 82.6%

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

   5. Applied egg-rr 82.6%

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

      1. fma-udef 82.6%

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

   7. Applied egg-rr 82.6%

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

      1. cancel-sign-sub-inv 82.6%

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

      2. associate-*r* 82.6%

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

      3. *-commutative 82.6%

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

      4. associate-*r* 82.6%

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

      5. count-2 82.6%

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

      6. associate-*r* 82.6%

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

      7. cancel-sign-sub-inv 82.6%

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

      8. *-commutative 82.6%

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

      9. associate-*r* 82.6%

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

      10. distribute-rgt-out 91.2%

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

      11. +-commutative 91.2%

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

      12. distribute-lft-neg-in 91.2%

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

      13. metadata-eval 91.2%

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

   9. Simplified 91.2%

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

   10. Taylor expanded in x.re around inf 72.6%

       \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 72.6%

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

   12. Simplified 72.6%

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

   if -7.00000000000000041e-86 < x.re < 4.8e14

   1. Initial program 88.3%

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

      1. sqr-neg 88.3%

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

      2. difference-of-squares 88.3%

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

      3. sub-neg 88.3%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. fma-def 99.7%

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

      3. *-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x.re around 0 82.1%

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

      1. unpow2 82.1%

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

      2. unpow2 82.1%

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

      3. distribute-rgt-out 82.1%

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

      4. metadata-eval 82.1%

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

      5. associate-*r* 82.1%

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

      6. *-commutative 82.1%

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

      7. associate-*l* 93.5%

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

      8. *-commutative 93.5%

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

   8. Simplified 93.5%

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

   if 4.8e14 < x.re

   1. Initial program 82.6%

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

      1. sqr-neg 82.6%

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

      2. difference-of-squares 92.7%

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

      3. sub-neg 92.7%

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

      4. associate-*l* 92.7%

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

      5. sub-neg 92.7%

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

      6. remove-double-neg 92.7%

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

      7. +-commutative 92.7%

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

      8. *-commutative 92.7%

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

      9. *-commutative 92.7%

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

      10. distribute-rgt-out 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in x.re around inf 82.0%

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

      1. unpow2 82.0%

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

   6. Simplified 82.0%

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

   7. Taylor expanded in x.re around inf 62.5%

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

      1. unpow3 62.5%

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

      2. mul-1-neg 62.5%

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

      3. unpow2 62.5%

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

      4. distribute-rgt-neg-in 62.5%

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

      5. distribute-lft-in 84.3%

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

      6. sub-neg 84.3%

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

      7. *-commutative 84.3%

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

   9. Simplified 84.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -7 \cdot 10^{-86}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \]

Alternative 6: 77.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.75e-12)
   (* x.re (* x.re x.re))
   (* -3.0 (* x.re (* x.im x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.75e-12) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= 1.75d-12) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (-3.0d0) * (x_46re * (x_46im * x_46im))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.75e-12) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.75e-12:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.75e-12)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.75e-12)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 1.75e-12], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.75e-12

   1. Initial program 86.9%

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

      1. sqr-neg 86.9%

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

      2. difference-of-squares 90.5%

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

      3. sub-neg 90.5%

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

      4. associate-*l* 94.2%

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

      5. sub-neg 94.2%

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

      6. remove-double-neg 94.2%

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

      7. +-commutative 94.2%

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

      8. *-commutative 94.2%

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

      9. *-commutative 94.2%

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

      10. distribute-rgt-out 94.2%

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

   3. Simplified 94.2%

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

      1. cancel-sign-sub-inv 94.2%

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

      2. fma-def 94.2%

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

      3. *-commutative 94.2%

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

   5. Applied egg-rr 94.2%

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

      1. fma-udef 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. cancel-sign-sub-inv 94.2%

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

      2. associate-*r* 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-*r* 90.5%

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

      5. count-2 90.5%

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

      6. associate-*r* 90.5%

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

      7. cancel-sign-sub-inv 90.5%

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

      8. *-commutative 90.5%

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

      9. associate-*r* 90.5%

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

      10. distribute-rgt-out 94.5%

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

      11. +-commutative 94.5%

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

      12. distribute-lft-neg-in 94.5%

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

      13. metadata-eval 94.5%

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

   9. Simplified 94.5%

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

   10. Taylor expanded in x.re around inf 69.2%

       \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 69.2%

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

   12. Simplified 69.2%

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

   if 1.75e-12 < x.im

   1. Initial program 70.2%

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

      1. sqr-neg 70.2%

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

      2. difference-of-squares 78.6%

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

      3. sub-neg 78.6%

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

      4. associate-*l* 86.4%

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

      5. sub-neg 86.4%

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

      6. remove-double-neg 86.4%

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

      7. +-commutative 86.4%

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

      8. *-commutative 86.4%

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

      9. *-commutative 86.4%

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

      10. distribute-rgt-out 86.4%

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

   3. Simplified 86.4%

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

      1. cancel-sign-sub-inv 86.4%

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

      2. fma-def 86.5%

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

      3. *-commutative 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. fma-udef 86.4%

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

   7. Applied egg-rr 86.4%

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

      1. cancel-sign-sub-inv 86.4%

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

      2. associate-*r* 86.4%

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

      3. *-commutative 86.4%

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

      4. associate-*r* 78.6%

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

      5. count-2 78.6%

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

      6. associate-*r* 78.6%

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

      7. cancel-sign-sub-inv 78.6%

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

      8. *-commutative 78.6%

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

      9. associate-*r* 78.5%

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

      10. distribute-rgt-out 80.3%

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

      11. +-commutative 80.3%

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

      12. distribute-lft-neg-in 80.3%

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

      13. metadata-eval 80.3%

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

   9. Simplified 80.3%

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

       1. distribute-lft-in 78.5%

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

       2. *-commutative 78.5%

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

       3. +-commutative 78.5%

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

       4. difference-of-squares 70.2%

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

       5. associate-*r* 70.2%

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

   11. Applied egg-rr 70.2%

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

       1. associate-*l* 70.2%

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

       2. distribute-lft-in 71.9%

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

       3. unpow2 71.9%

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

       4. sub-neg 71.9%

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

       5. mul-1-neg 71.9%

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

       6. unpow2 71.9%

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

       7. associate-+r+ 71.9%

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

       8. unpow2 71.9%

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

       9. distribute-rgt-out 71.9%

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

       10. metadata-eval 71.9%

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

       11. *-commutative 71.9%

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

       12. unpow2 71.9%

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

       13. unpow2 71.9%

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

       14. unpow2 71.9%

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

   13. Simplified 71.9%

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

   14. Taylor expanded in x.re around 0 66.9%

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

       1. unpow2 66.9%

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

   16. Simplified 66.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \]

Alternative 7: 83.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.75e-12)
   (* x.re (* x.re x.re))
   (* x.im (* (* x.re x.im) -3.0))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.75e-12) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= 1.75d-12) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46im * ((x_46re * x_46im) * (-3.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.75e-12) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.75e-12:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.75e-12)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * -3.0));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.75e-12)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 1.75e-12], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.75e-12

   1. Initial program 86.9%

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

      1. sqr-neg 86.9%

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

      2. difference-of-squares 90.5%

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

      3. sub-neg 90.5%

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

      4. associate-*l* 94.2%

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

      5. sub-neg 94.2%

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

      6. remove-double-neg 94.2%

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

      7. +-commutative 94.2%

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

      8. *-commutative 94.2%

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

      9. *-commutative 94.2%

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

      10. distribute-rgt-out 94.2%

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

   3. Simplified 94.2%

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

      1. cancel-sign-sub-inv 94.2%

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

      2. fma-def 94.2%

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

      3. *-commutative 94.2%

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

   5. Applied egg-rr 94.2%

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

      1. fma-udef 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. cancel-sign-sub-inv 94.2%

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

      2. associate-*r* 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-*r* 90.5%

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

      5. count-2 90.5%

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

      6. associate-*r* 90.5%

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

      7. cancel-sign-sub-inv 90.5%

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

      8. *-commutative 90.5%

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

      9. associate-*r* 90.5%

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

      10. distribute-rgt-out 94.5%

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

      11. +-commutative 94.5%

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

      12. distribute-lft-neg-in 94.5%

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

      13. metadata-eval 94.5%

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

   9. Simplified 94.5%

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

   10. Taylor expanded in x.re around inf 69.2%

       \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 69.2%

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

   12. Simplified 69.2%

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

   if 1.75e-12 < x.im

   1. Initial program 70.2%

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

      1. sqr-neg 70.2%

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

      2. difference-of-squares 78.6%

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

      3. sub-neg 78.6%

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

      4. associate-*l* 86.4%

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

      5. sub-neg 86.4%

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

      6. remove-double-neg 86.4%

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

      7. +-commutative 86.4%

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

      8. *-commutative 86.4%

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

      9. *-commutative 86.4%

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

      10. distribute-rgt-out 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x.re around 0 66.9%

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

      1. distribute-rgt-out-- 66.9%

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

      2. metadata-eval 66.9%

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

      3. unpow2 66.9%

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

      4. associate-*r* 66.9%

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

      5. *-commutative 66.9%

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

      6. associate-*l* 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in x.im around 0 74.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \]

Alternative 8: 83.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.32 \cdot 10^{-12}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.32e-12)
   (* x.re (* x.re x.re))
   (* x.im (* x.re (* x.im -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.32e-12) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= 1.32d-12) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46im * (x_46re * (x_46im * (-3.0d0)))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.32e-12) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.32e-12:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.32e-12)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.32e-12)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 1.32e-12], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.32e-12

   1. Initial program 86.9%

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

      1. sqr-neg 86.9%

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

      2. difference-of-squares 90.5%

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

      3. sub-neg 90.5%

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

      4. associate-*l* 94.2%

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

      5. sub-neg 94.2%

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

      6. remove-double-neg 94.2%

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

      7. +-commutative 94.2%

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

      8. *-commutative 94.2%

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

      9. *-commutative 94.2%

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

      10. distribute-rgt-out 94.2%

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

   3. Simplified 94.2%

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

      1. cancel-sign-sub-inv 94.2%

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

      2. fma-def 94.2%

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

      3. *-commutative 94.2%

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

   5. Applied egg-rr 94.2%

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

      1. fma-udef 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. cancel-sign-sub-inv 94.2%

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

      2. associate-*r* 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-*r* 90.5%

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

      5. count-2 90.5%

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

      6. associate-*r* 90.5%

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

      7. cancel-sign-sub-inv 90.5%

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

      8. *-commutative 90.5%

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

      9. associate-*r* 90.5%

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

      10. distribute-rgt-out 94.5%

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

      11. +-commutative 94.5%

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

      12. distribute-lft-neg-in 94.5%

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

      13. metadata-eval 94.5%

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

   9. Simplified 94.5%

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

   10. Taylor expanded in x.re around inf 69.2%

       \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 69.2%

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

   12. Simplified 69.2%

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

   if 1.32e-12 < x.im

   1. Initial program 70.2%

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

      1. sqr-neg 70.2%

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

      2. difference-of-squares 78.6%

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

      3. sub-neg 78.6%

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

      4. associate-*l* 86.4%

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

      5. sub-neg 86.4%

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

      6. remove-double-neg 86.4%

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

      7. +-commutative 86.4%

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

      8. *-commutative 86.4%

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

      9. *-commutative 86.4%

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

      10. distribute-rgt-out 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x.re around 0 66.9%

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

      1. distribute-rgt-out-- 66.9%

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

      2. metadata-eval 66.9%

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

      3. unpow2 66.9%

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

      4. associate-*r* 66.9%

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

      5. *-commutative 66.9%

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

      6. associate-*l* 74.8%

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

   6. Simplified 74.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.32 \cdot 10^{-12}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 9: 83.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 8.4 \cdot 10^{-13}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 8.4e-13) (* x.re (* x.re x.re)) (* (* x.re x.im) (* x.im -3.0))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 8.4e-13) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= 8.4d-13) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (x_46re * x_46im) * (x_46im * (-3.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 8.4e-13) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 8.4e-13:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 8.4e-13)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_im * -3.0));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 8.4e-13)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 8.4e-13], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 8.39999999999999955e-13

   1. Initial program 86.9%

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

      1. sqr-neg 86.9%

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

      2. difference-of-squares 90.5%

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

      3. sub-neg 90.5%

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

      4. associate-*l* 94.2%

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

      5. sub-neg 94.2%

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

      6. remove-double-neg 94.2%

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

      7. +-commutative 94.2%

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

      8. *-commutative 94.2%

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

      9. *-commutative 94.2%

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

      10. distribute-rgt-out 94.2%

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

   3. Simplified 94.2%

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

      1. cancel-sign-sub-inv 94.2%

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

      2. fma-def 94.2%

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

      3. *-commutative 94.2%

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

   5. Applied egg-rr 94.2%

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

      1. fma-udef 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. cancel-sign-sub-inv 94.2%

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

      2. associate-*r* 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-*r* 90.5%

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

      5. count-2 90.5%

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

      6. associate-*r* 90.5%

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

      7. cancel-sign-sub-inv 90.5%

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

      8. *-commutative 90.5%

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

      9. associate-*r* 90.5%

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

      10. distribute-rgt-out 94.5%

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

      11. +-commutative 94.5%

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

      12. distribute-lft-neg-in 94.5%

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

      13. metadata-eval 94.5%

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

   9. Simplified 94.5%

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

   10. Taylor expanded in x.re around inf 69.2%

       \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 69.2%

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

   12. Simplified 69.2%

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

   if 8.39999999999999955e-13 < x.im

   1. Initial program 70.2%

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

      1. sqr-neg 70.2%

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

      2. difference-of-squares 78.6%

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

      3. sub-neg 78.6%

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

      4. associate-*l* 86.4%

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

      5. sub-neg 86.4%

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

      6. remove-double-neg 86.4%

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

      7. +-commutative 86.4%

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

      8. *-commutative 86.4%

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

      9. *-commutative 86.4%

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

      10. distribute-rgt-out 86.4%

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

   3. Simplified 86.4%

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

      1. cancel-sign-sub-inv 86.4%

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

      2. fma-def 86.5%

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

      3. *-commutative 86.5%

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

   5. Applied egg-rr 86.5%

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

   6. Taylor expanded in x.re around 0 66.9%

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

      1. unpow2 66.9%

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

      2. unpow2 66.9%

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

      3. distribute-rgt-out 66.9%

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

      4. metadata-eval 66.9%

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

      5. associate-*r* 66.9%

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

      6. *-commutative 66.9%

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

      7. associate-*l* 74.8%

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

      8. *-commutative 74.8%

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

   8. Simplified 74.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 8.4 \cdot 10^{-13}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \]

Alternative 10: 61.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.2 \cdot 10^{+180}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 2.2e+180) (* x.re (* x.re x.re)) (* (* x.re x.re) x.im)))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.2e+180) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= 2.2d+180) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.2e+180) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 2.2e+180:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.2e+180)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 2.2e+180)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 2.2e+180], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.1999999999999999e180

   1. Initial program 84.8%

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

      1. sqr-neg 84.8%

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

      2. difference-of-squares 88.2%

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

      3. sub-neg 88.2%

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

      4. associate-*l* 93.0%

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

      5. sub-neg 93.0%

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

      6. remove-double-neg 93.0%

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

      7. +-commutative 93.0%

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

      8. *-commutative 93.0%

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

      9. *-commutative 93.0%

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

      10. distribute-rgt-out 93.0%

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

   3. Simplified 93.0%

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

      1. cancel-sign-sub-inv 93.0%

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

      2. fma-def 93.0%

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

      3. *-commutative 93.0%

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

   5. Applied egg-rr 93.0%

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

      1. fma-udef 93.0%

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

   7. Applied egg-rr 93.0%

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

      1. cancel-sign-sub-inv 93.0%

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

      2. associate-*r* 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-*r* 88.2%

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

      5. count-2 88.2%

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

      6. associate-*r* 88.2%

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

      7. cancel-sign-sub-inv 88.2%

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

      8. *-commutative 88.2%

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

      9. associate-*r* 88.2%

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

      10. distribute-rgt-out 92.0%

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

      11. +-commutative 92.0%

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

      12. distribute-lft-neg-in 92.0%

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

      13. metadata-eval 92.0%

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

   9. Simplified 92.0%

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

   10. Taylor expanded in x.re around inf 63.4%

       \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 63.4%

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

   12. Simplified 63.4%

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

   if 2.1999999999999999e180 < x.im

   1. Initial program 63.8%

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

      1. sqr-neg 63.8%

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

      2. difference-of-squares 82.0%

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

      3. sub-neg 82.0%

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

      4. associate-*l* 86.4%

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

      5. sub-neg 86.4%

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

      6. remove-double-neg 86.4%

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

      7. +-commutative 86.4%

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

      8. *-commutative 86.4%

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

      9. *-commutative 86.4%

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

      10. distribute-rgt-out 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x.re around inf 64.6%

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

      1. unpow2 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in x.re around inf 27.7%

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

      1. unpow3 27.7%

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

      2. mul-1-neg 27.7%

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

      3. unpow2 27.7%

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

      4. distribute-rgt-neg-in 27.7%

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

      5. distribute-lft-in 45.9%

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

      6. sub-neg 45.9%

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

      7. *-commutative 45.9%

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

   9. Simplified 45.9%

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

   10. Taylor expanded in x.re around 0 41.3%

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

       1. unpow2 41.3%

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

       2. associate-*r* 41.3%

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

       3. mul-1-neg 41.3%

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

   12. Simplified 41.3%

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

       1. expm1-log1p-u 0.2%

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

       2. expm1-udef 0.3%

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

       3. add-sqr-sqrt 0.2%

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

       4. sqrt-unprod 23.9%

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

       5. sqr-neg 23.9%

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

       6. sqrt-unprod 23.9%

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

       7. add-sqr-sqrt 23.9%

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

       8. associate-*l* 23.9%

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

   14. Applied egg-rr 23.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im\right)\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 24.0%

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

       2. expm1-log1p 24.0%

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

       3. associate-*r* 24.0%

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

       4. *-commutative 24.0%

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

   16. Simplified 24.0%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.2 \cdot 10^{+180}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 11: 30.6% accurate, 3.8× speedup

Math

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

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 (* (* x.re x.re) x.im))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	return (x_46_re * x_46_re) * x_46_im;
}

Fortran

NOTE: x.im should be positive before calling this function
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
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	return (x_46_re * x_46_re) * x_46_im;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	return (x_46_re * x_46_re) * x_46_im

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	return Float64(Float64(x_46_re * x_46_re) * x_46_im)
end

MATLAB

x.im = abs(x.im)
function tmp = code(x_46_re, x_46_im)
	tmp = (x_46_re * x_46_re) * x_46_im;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

   1. sqr-neg 83.0%

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

   2. difference-of-squares 87.7%

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

   3. sub-neg 87.7%

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

   4. associate-*l* 92.4%

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

   5. sub-neg 92.4%

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

   6. remove-double-neg 92.4%

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

   7. +-commutative 92.4%

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

   8. *-commutative 92.4%

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

   9. *-commutative 92.4%

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

   10. distribute-rgt-out 92.4%

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

3. Simplified 92.4%

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

4. Taylor expanded in x.re around inf 70.7%

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

   1. unpow2 70.7%

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

6. Simplified 70.7%

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

7. Taylor expanded in x.re around inf 50.3%

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

   1. unpow3 50.2%

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

   2. mul-1-neg 50.2%

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

   3. unpow2 50.2%

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

   4. distribute-rgt-neg-in 50.2%

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

   5. distribute-lft-in 65.9%

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

   6. sub-neg 65.9%

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

   7. *-commutative 65.9%

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

9. Simplified 65.9%

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

10. Taylor expanded in x.re around 0 34.6%

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

    1. unpow2 34.6%

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

    2. associate-*r* 34.6%

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

    3. mul-1-neg 34.6%

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

12. Simplified 34.6%

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

    1. expm1-log1p-u 25.3%

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

    2. expm1-udef 25.2%

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

    3. add-sqr-sqrt 10.1%

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

    4. sqrt-unprod 22.4%

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

    5. sqr-neg 22.4%

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

    6. sqrt-unprod 20.5%

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

    7. add-sqr-sqrt 20.5%

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

    8. associate-*l* 20.2%

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

14. Applied egg-rr 20.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im\right)\right)} - 1} \]
15. Step-by-step derivation

    1. expm1-def 20.2%

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

    2. expm1-log1p 28.1%

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

    3. associate-*r* 29.9%

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

    4. *-commutative 29.9%

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

16. Simplified 29.9%

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

17. Final simplification 29.9%

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

Developer target: 87.7% 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 2023279 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* 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)))