math.cube on complex, real part

Percentage Accurate: 82.8% → 96.8%

Time: 6.9s

Alternatives: 7

Speedup: 1.1×

• 43.8% 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: 96.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re, 3.0], $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 92.8%

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

   3. Simplified 91.1%

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

      1. unpow3 91.0%

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

      2. fma-def 92.8%

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

      3. associate-*r* 99.8%

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

   5. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x.re around inf 72.4%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

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

Alternative 2: 96.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = pow(x_46_re, 3.0);
	}
	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_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = Math.pow(x_46_re, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= math.inf:
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	else:
		tmp = math.pow(x_46_re, 3.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re, 3.0], $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 92.8%

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

      1. sqr-neg 92.8%

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

      2. difference-of-squares 92.8%

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

      3. sub-neg 92.8%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. fma-def 99.8%

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

      3. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. add-cube-cbrt 99.2%

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

      3. pow3 99.2%

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

      4. *-commutative 99.2%

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

   7. Applied egg-rr 99.2%

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

      1. fma-udef 99.2%

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

      2. unpow3 99.2%

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

      3. add-cube-cbrt 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. distribute-lft-neg-out 99.7%

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

       2. unsub-neg 99.7%

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

       3. associate-*r* 99.7%

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

       4. *-commutative 99.7%

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

       5. *-commutative 99.7%

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

   11. Simplified 99.7%

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

   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%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x.re around inf 72.4%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

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

Alternative 3: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.9 \cdot 10^{-94} \lor \neg \left(x.re \leq 2.45 \cdot 10^{-67}\right):\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -1.9e-94) (not (<= x.re 2.45e-67)))
   (- (* (- x.re x.im) (* x.re (+ x.re x.im))) (* x.re (* x.im x.im)))
   (* x.im (* (* x.re x.im) -3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.9e-94) || !(x_46_re <= 2.45e-67)) {
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-1.9d-94)) .or. (.not. (x_46re <= 2.45d-67))) then
        tmp = ((x_46re - x_46im) * (x_46re * (x_46re + x_46im))) - (x_46re * (x_46im * x_46im))
    else
        tmp = x_46im * ((x_46re * x_46im) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.9e-94) || !(x_46_re <= 2.45e-67)) {
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -1.9e-94) or not (x_46_re <= 2.45e-67):
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_re * (x_46_im * x_46_im))
	else:
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -1.9e-94) || !(x_46_re <= 2.45e-67))
		tmp = Float64(Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re * Float64(x_46_re + x_46_im))) - Float64(x_46_re * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -1.9e-94) || ~((x_46_re <= 2.45e-67)))
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_re * (x_46_im * x_46_im));
	else
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -1.9e-94], N[Not[LessEqual[x$46$re, 2.45e-67]], $MachinePrecision]], N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.9e-94 or 2.44999999999999997e-67 < x.re

   1. Initial program 81.5%

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

      1. sqr-neg 81.5%

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

      2. difference-of-squares 86.4%

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

      3. sub-neg 86.4%

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

      4. associate-*l* 86.9%

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

      5. sub-neg 86.9%

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

      6. remove-double-neg 86.9%

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

      7. +-commutative 86.9%

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

      8. *-commutative 86.9%

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

      9. *-commutative 86.9%

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

      10. distribute-rgt-out 86.9%

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

   3. Simplified 86.9%

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

      1. distribute-lft-in 86.9%

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

      2. add-cube-cbrt 86.9%

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

      3. fma-def 86.9%

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

      4. pow2 86.9%

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

   5. Applied egg-rr 86.8%

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

   6. Taylor expanded in x.im around 0 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

   8. Simplified 77.2%

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

   if -1.9e-94 < x.re < 2.44999999999999997e-67

   1. Initial program 83.6%

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

      1. sqr-neg 83.6%

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

      2. difference-of-squares 83.6%

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

      3. sub-neg 83.6%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. fma-def 99.7%

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

      3. *-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x.re around 0 81.9%

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

      1. distribute-rgt-out 81.9%

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

      2. unpow2 81.9%

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

      3. metadata-eval 81.9%

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

   8. Simplified 81.9%

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

   9. Taylor expanded in x.re around 0 81.8%

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

       1. *-commutative 81.8%

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

       2. associate-*r* 81.9%

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

       3. unpow2 81.9%

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

       4. associate-*l* 97.9%

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

       5. *-commutative 97.9%

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

       6. associate-*r* 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in x.im around 0 98.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.9 \cdot 10^{-94} \lor \neg \left(x.re \leq 2.45 \cdot 10^{-67}\right):\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \]

Alternative 4: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 8 \cdot 10^{+87}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.im \cdot x.im\right) \cdot \left(x.re \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 8e+87)
   (+ (* x.re (- (* x.re x.re) (* x.im x.im))) (* (* x.im x.im) (* x.re -2.0)))
   (* x.im (* x.re (* x.im -3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 8e+87) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + ((x_46_im * x_46_im) * (x_46_re * -2.0));
	} else {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 8d+87) then
        tmp = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) + ((x_46im * x_46im) * (x_46re * (-2.0d0)))
    else
        tmp = x_46im * (x_46re * (x_46im * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 8e+87) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + ((x_46_im * x_46_im) * (x_46_re * -2.0));
	} else {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 8e+87:
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + ((x_46_im * x_46_im) * (x_46_re * -2.0))
	else:
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 8e+87)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(Float64(x_46_im * x_46_im) * Float64(x_46_re * -2.0)));
	else
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 8e+87)
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + ((x_46_im * x_46_im) * (x_46_re * -2.0));
	else
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 8e+87], N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(x$46$re * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.9999999999999997e87

   1. Initial program 87.8%

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

      1. sqr-neg 87.8%

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

      2. *-commutative 87.8%

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

      3. fma-neg 87.8%

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

      4. sqr-neg 87.8%

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

      5. +-commutative 87.8%

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

      6. *-commutative 87.8%

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

      7. *-commutative 87.8%

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

      8. distribute-lft-out 87.8%

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

      9. associate-*r* 87.8%

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

      10. distribute-rgt-neg-in 87.8%

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

      11. distribute-neg-out 87.8%

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

      12. neg-mul-1 87.8%

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

      13. neg-mul-1 87.8%

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

      14. distribute-rgt-out 87.8%

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

      15. metadata-eval 87.8%

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

   3. Simplified 87.8%

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

      1. fma-udef 87.8%

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

   5. Applied egg-rr 87.8%

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

   if 7.9999999999999997e87 < x.im

   1. Initial program 55.4%

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

      1. sqr-neg 55.4%

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

      2. difference-of-squares 64.5%

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

      3. sub-neg 64.5%

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

      4. associate-*l* 81.4%

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

      5. sub-neg 81.4%

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

      6. remove-double-neg 81.4%

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

      7. +-commutative 81.4%

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

      8. *-commutative 81.4%

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

      9. *-commutative 81.4%

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

      10. distribute-rgt-out 81.4%

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

   3. Simplified 81.4%

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

      1. cancel-sign-sub-inv 81.4%

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

      2. fma-def 81.6%

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

      3. *-commutative 81.6%

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

   5. Applied egg-rr 81.6%

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

   6. Taylor expanded in x.re around 0 64.5%

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

      1. distribute-rgt-out 64.5%

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

      2. unpow2 64.5%

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

      3. metadata-eval 64.5%

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

   8. Simplified 64.5%

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

   9. Taylor expanded in x.re around 0 64.5%

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

       1. *-commutative 64.5%

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

       2. associate-*r* 64.5%

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

       3. unpow2 64.5%

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

       4. associate-*l* 81.6%

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

       5. *-commutative 81.6%

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

       6. associate-*r* 81.6%

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

   11. Simplified 81.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 8 \cdot 10^{+87}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.im \cdot x.im\right) \cdot \left(x.re \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 5: 91.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

   1. sqr-neg 82.2%

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

   2. difference-of-squares 85.4%

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

   3. sub-neg 85.4%

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

   4. associate-*l* 91.5%

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

   5. sub-neg 91.5%

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

   6. remove-double-neg 91.5%

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

   7. +-commutative 91.5%

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

   8. *-commutative 91.5%

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

   9. *-commutative 91.5%

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

   10. distribute-rgt-out 91.5%

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

3. Simplified 91.5%

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

   1. cancel-sign-sub-inv 91.5%

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

   2. fma-def 91.6%

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

   3. *-commutative 91.6%

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

5. Applied egg-rr 91.6%

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

   1. *-commutative 91.6%

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

   2. add-cube-cbrt 91.1%

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

   3. pow3 91.1%

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

   4. *-commutative 91.1%

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

7. Applied egg-rr 91.1%

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

   1. fma-udef 91.1%

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

   2. unpow3 91.1%

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

   3. add-cube-cbrt 91.5%

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

9. Applied egg-rr 91.5%

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

    1. distribute-lft-neg-out 91.5%

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

    2. unsub-neg 91.5%

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

    3. associate-*r* 91.5%

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

    4. *-commutative 91.5%

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

    5. *-commutative 91.5%

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

11. Simplified 91.5%

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

12. Final simplification 91.5%

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

Alternative 6: 36.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return -2.0 * (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 = (-2.0d0) * (x_46im * (x_46re * x_46im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

   1. sqr-neg 82.2%

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

   2. difference-of-squares 85.4%

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

   3. sub-neg 85.4%

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

   4. associate-*l* 91.5%

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

   5. sub-neg 91.5%

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

   6. remove-double-neg 91.5%

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

   7. +-commutative 91.5%

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

   8. *-commutative 91.5%

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

   9. *-commutative 91.5%

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

   10. distribute-rgt-out 91.5%

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

3. Simplified 91.5%

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

   1. cancel-sign-sub-inv 91.5%

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

   2. fma-def 91.6%

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

   3. *-commutative 91.6%

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

5. Applied egg-rr 91.6%

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

   1. distribute-lft-in 91.6%

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

   2. add-cube-cbrt 91.4%

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

   3. fma-def 91.4%

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

   4. pow2 91.4%

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

7. Applied egg-rr 91.4%

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

8. Taylor expanded in x.re around 0 31.6%

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

   1. unpow2 31.6%

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

   2. associate-*l* 32.5%

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

10. Simplified 32.5%

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

11. Final simplification 32.5%

    \[\leadsto -2 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right) \]

Alternative 7: 56.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

   1. sqr-neg 82.2%

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

   2. difference-of-squares 85.4%

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

   3. sub-neg 85.4%

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

   4. associate-*l* 91.5%

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

   5. sub-neg 91.5%

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

   6. remove-double-neg 91.5%

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

   7. +-commutative 91.5%

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

   8. *-commutative 91.5%

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

   9. *-commutative 91.5%

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

   10. distribute-rgt-out 91.5%

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

3. Simplified 91.5%

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

   1. cancel-sign-sub-inv 91.5%

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

   2. fma-def 91.6%

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

   3. *-commutative 91.6%

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

5. Applied egg-rr 91.6%

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

6. Taylor expanded in x.re around 0 47.4%

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

   1. distribute-rgt-out 47.4%

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

   2. unpow2 47.4%

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

   3. metadata-eval 47.4%

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

8. Simplified 47.4%

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

9. Taylor expanded in x.re around 0 47.3%

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

    1. *-commutative 47.3%

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

    2. associate-*r* 47.4%

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

    3. unpow2 47.4%

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

    4. associate-*l* 53.5%

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

    5. *-commutative 53.5%

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

    6. associate-*r* 53.6%

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

11. Simplified 53.6%

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

12. Taylor expanded in x.im around 0 53.6%

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

13. Final simplification 53.6%

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

Developer target: 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 2023271 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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