math.cube on complex, real part

Percentage Accurate: 82.8% → 99.8%

Time: 9.2s

Alternatives: 8

Speedup: 0.6×

• 44.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.re\right) - \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x.re}{x.im}, \frac{x.re}{x.im}, -3\right) \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im x.im)) x.re)
       (* (+ (* x.im x.re) (* x.im x.re)) x.im))
      INFINITY)
   (- (* (- x.re x.im) (* (+ x.im x.re) x.re)) (* (* (+ x.im x.im) x.re) x.im))
   (* (* (fma (/ x.re x.im) (/ x.re x.im) -3.0) x.re) (* x.im x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((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_im + x_46_re) * x_46_re)) - (((x_46_im + x_46_im) * x_46_re) * x_46_im);
	} else {
		tmp = (fma((x_46_re / x_46_im), (x_46_re / x_46_im), -3.0) * x_46_re) * (x_46_im * x_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (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_im * x_46_re) + Float64(x_46_im * x_46_re)) * x_46_im)) <= Inf)
		tmp = Float64(Float64(Float64(x_46_re - x_46_im) * Float64(Float64(x_46_im + x_46_re) * x_46_re)) - Float64(Float64(Float64(x_46_im + x_46_im) * x_46_re) * x_46_im));
	else
		tmp = Float64(Float64(fma(Float64(x_46_re / x_46_im), Float64(x_46_re / x_46_im), -3.0) * x_46_re) * Float64(x_46_im * x_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[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$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(N[(x$46$im + x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$46$re / x$46$im), $MachinePrecision] * N[(x$46$re / x$46$im), $MachinePrecision] + -3.0), $MachinePrecision] * x$46$re), $MachinePrecision] * N[(x$46$im * 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 92.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 14.8

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

   4. Applied rewrites 14.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift--.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 14.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 14.8

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

   6. Applied rewrites 14.8%

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

   7. Taylor expanded in x.im around inf

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

   9. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.re\right) - \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x.re}{x.im}, \frac{x.re}{x.im}, -3\right) \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((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_im + x_46_re) * x_46_re)) - (((x_46_im + x_46_im) * x_46_re) * x_46_im);
	} else {
		tmp = (x_46_im * x_46_im) * x_46_re;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((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_im + x_46_re) * x_46_re)) - (((x_46_im + x_46_im) * x_46_re) * x_46_im);
	} else {
		tmp = (x_46_im * x_46_im) * x_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((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_im + x_46_re) * x_46_re)) - (((x_46_im + x_46_im) * x_46_re) * x_46_im)
	else:
		tmp = (x_46_im * x_46_im) * x_46_re
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((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_im + x_46_re) * x_46_re)) - (((x_46_im + x_46_im) * x_46_re) * x_46_im);
	else
		tmp = (x_46_im * x_46_im) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[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$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(N[(x$46$im + x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$re), $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 92.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   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. Add Preprocessing

   3. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. distribute-neg-frac N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. distribute-neg-in N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-neg.f64 85.2

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in x.im around inf

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

      1. distribute-rgt-out-- N/A

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

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

         \[\leadsto {x.im}^{2} \cdot \color{blue}{x.re} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x.im}^{2} \cdot x.re} \]

      5. unpow2 N/A

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

      6. lower-*.f64 85.2

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

   10. Applied rewrites 85.2%

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

4. Final simplification 98.2%

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

Alternative 3: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.re, x.re, \left(\left(-3 \cdot x.re\right) \cdot x.im\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im x.im)) x.re)
       (* (+ (* x.im x.re) (* x.im x.re)) x.im))
      INFINITY)
   (fma (* x.re x.re) x.re (* (* (* -3.0 x.re) x.im) x.im))
   (* (* x.im x.im) x.re)))

C

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

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (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_im * x_46_re) + Float64(x_46_im * x_46_re)) * x_46_im)) <= Inf)
		tmp = fma(Float64(x_46_re * x_46_re), x_46_re, Float64(Float64(Float64(-3.0 * x_46_re) * x_46_im) * x_46_im));
	else
		tmp = Float64(Float64(x_46_im * x_46_im) * x_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[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$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re + N[(N[(N[(-3.0 * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$re), $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 92.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. pow-prod-up N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. metadata-eval N/A

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

   4. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-out-- N/A

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

      13. distribute-rgt-in N/A

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

      14. associate--l+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 92.8%

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

   9. Applied rewrites 99.7%

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

   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. Add Preprocessing

   3. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. distribute-neg-frac N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. distribute-neg-in N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-neg.f64 85.2

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in x.im around inf

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

      1. distribute-rgt-out-- N/A

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

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

         \[\leadsto {x.im}^{2} \cdot \color{blue}{x.re} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x.im}^{2} \cdot x.re} \]

      5. unpow2 N/A

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

      6. lower-*.f64 85.2

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

   10. Applied rewrites 85.2%

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

4. Final simplification 98.1%

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

Alternative 4: 60.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\left(-3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im x.im)) x.re)
       (* (+ (* x.im x.re) (* x.im x.re)) x.im))
      -5e-324)
   (* (* -3.0 (* x.im x.re)) x.im)
   (* (* x.re x.re) x.re)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324) {
		tmp = (-3.0 * (x_46_im * x_46_re)) * x_46_im;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im * x_46im)) * x_46re) - (((x_46im * x_46re) + (x_46im * x_46re)) * x_46im)) <= (-5d-324)) then
        tmp = ((-3.0d0) * (x_46im * x_46re)) * x_46im
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324) {
		tmp = (-3.0 * (x_46_im * x_46_re)) * x_46_im;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324:
		tmp = (-3.0 * (x_46_im * x_46_re)) * x_46_im
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (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_im * x_46_re) + Float64(x_46_im * x_46_re)) * x_46_im)) <= -5e-324)
		tmp = Float64(Float64(-3.0 * Float64(x_46_im * x_46_re)) * x_46_im);
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324)
		tmp = (-3.0 * (x_46_im * x_46_re)) * x_46_im;
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[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$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -5e-324], N[(N[(-3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.94066e-324

   1. Initial program 93.9%

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

   3. Taylor expanded in x.im around inf

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

      1. distribute-rgt-out-- N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 57.9%

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

   if -4.94066e-324 < (-.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 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. pow-prod-up N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. metadata-eval N/A

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

   4. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-out-- N/A

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

      13. distribute-rgt-in N/A

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

      14. associate--l+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 85.3%

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

   8. Taylor expanded in x.im around 0

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

   10. Applied rewrites 60.5%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\left(-3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im x.im)) x.re)
       (* (+ (* x.im x.re) (* x.im x.re)) x.im))
      -5e-324)
   (* (* (* x.im x.re) x.im) -3.0)
   (* (* x.re x.re) x.re)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324) {
		tmp = ((x_46_im * x_46_re) * x_46_im) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im * x_46im)) * x_46re) - (((x_46im * x_46re) + (x_46im * x_46re)) * x_46im)) <= (-5d-324)) then
        tmp = ((x_46im * x_46re) * x_46im) * (-3.0d0)
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324) {
		tmp = ((x_46_im * x_46_re) * x_46_im) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324:
		tmp = ((x_46_im * x_46_re) * x_46_im) * -3.0
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (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_im * x_46_re) + Float64(x_46_im * x_46_re)) * x_46_im)) <= -5e-324)
		tmp = Float64(Float64(Float64(x_46_im * x_46_re) * x_46_im) * -3.0);
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324)
		tmp = ((x_46_im * x_46_re) * x_46_im) * -3.0;
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[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$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -5e-324], N[(N[(N[(x$46$im * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.94066e-324

   1. Initial program 93.9%

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

   3. Taylor expanded in x.im around inf

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

      1. distribute-rgt-out-- N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 57.9%

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

   if -4.94066e-324 < (-.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 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. pow-prod-up N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. metadata-eval N/A

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

   4. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-out-- N/A

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

      13. distribute-rgt-in N/A

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

      14. associate--l+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 85.3%

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

   8. Taylor expanded in x.im around 0

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

   10. Applied rewrites 60.5%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im x.im)) x.re)
       (* (+ (* x.im x.re) (* x.im x.re)) x.im))
      -5e-324)
   (* (* (* x.im x.im) x.re) -3.0)
   (* (* x.re x.re) x.re)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324) {
		tmp = ((x_46_im * x_46_im) * x_46_re) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im * x_46im)) * x_46re) - (((x_46im * x_46re) + (x_46im * x_46re)) * x_46im)) <= (-5d-324)) then
        tmp = ((x_46im * x_46im) * x_46re) * (-3.0d0)
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324) {
		tmp = ((x_46_im * x_46_im) * x_46_re) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324:
		tmp = ((x_46_im * x_46_im) * x_46_re) * -3.0
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (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_im * x_46_re) + Float64(x_46_im * x_46_re)) * x_46_im)) <= -5e-324)
		tmp = Float64(Float64(Float64(x_46_im * x_46_im) * x_46_re) * -3.0);
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_im)) <= -5e-324)
		tmp = ((x_46_im * x_46_im) * x_46_re) * -3.0;
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[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$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -5e-324], N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.94066e-324

   1. Initial program 93.9%

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

   3. Taylor expanded in x.im around inf

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

      1. distribute-rgt-out-- N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   if -4.94066e-324 < (-.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 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. pow-prod-up N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. metadata-eval N/A

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

   4. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-out-- N/A

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

      13. distribute-rgt-in N/A

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

      14. associate--l+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 85.3%

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

   8. Taylor expanded in x.im around 0

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

   10. Applied rewrites 60.5%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval N/A

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

   14. pow2 N/A

       \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{\left(x.im \cdot x.im\right)}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\left({x.re}^{4} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{2}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   16. pow-prod-down N/A

       \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{2} \cdot {x.im}^{2}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   17. pow-prod-up N/A

       \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   18. lower-pow.f64 N/A

       \[\leadsto \frac{\left({x.re}^{4} - \color{blue}{{x.im}^{\left(2 + 2\right)}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   19. metadata-eval N/A

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

4. Applied rewrites 36.5%

   \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

5. Taylor expanded in x.im around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unpow3 N/A

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

   4. unpow2 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. distribute-rgt1-in N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-neg-in N/A

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

   9. metadata-eval N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. distribute-rgt-out-- N/A

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

   13. distribute-rgt-in N/A

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

   14. associate--l+ N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

7. Applied rewrites 88.5%

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

8. Taylor expanded in x.im around 0

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

10. Applied rewrites 54.1%

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

Alternative 8: 23.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Taylor expanded in x.im around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 61.9

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

5. Applied rewrites 61.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-inverses N/A

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

   9. distribute-neg-frac N/A

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

   10. +-inverses N/A

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

   11. +-inverses N/A

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

   12. flip-+ N/A

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

   13. distribute-neg-in N/A

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

   14. lower-+.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-neg.f64 53.6

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

7. Applied rewrites 53.6%

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

8. Taylor expanded in x.im around inf

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

   1. distribute-rgt-out-- N/A

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

   2. metadata-eval N/A

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

   3. *-rgt-identity N/A

      \[\leadsto {x.im}^{2} \cdot \color{blue}{x.re} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{{x.im}^{2} \cdot x.re} \]

   5. unpow2 N/A

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

   6. lower-*.f64 22.8

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

10. Applied rewrites 22.8%

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

Developer Target 1: 87.5% 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 2024250 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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