math.cube on complex, real part

Percentage Accurate: 83.1% → 99.8%

Time: 11.0s

Alternatives: 9

Speedup: 1.4×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.1% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\\ \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - t\_0 \leq \infty:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x.re}{x.im}, \frac{x.re}{x.im}, -3\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (+ (* x.re x.im) (* x.im x.re)) x.im)))
   (if (<= (- (* (- (* x.re x.re) (* x.im x.im)) x.re) t_0) INFINITY)
     (- (* (+ x.im x.re) (* (- x.re x.im) x.re)) t_0)
     (* (* (fma (/ x.re x.im) (/ x.re x.im) -3.0) (* x.im x.im)) x.re))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = ((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im;
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - t_0) <= ((double) INFINITY)) {
		tmp = ((x_46_im + x_46_re) * ((x_46_re - x_46_im) * x_46_re)) - t_0;
	} else {
		tmp = (fma((x_46_re / x_46_im), (x_46_re / x_46_im), -3.0) * (x_46_im * x_46_im)) * x_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * 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) - t_0) <= Inf)
		tmp = Float64(Float64(Float64(x_46_im + x_46_re) * Float64(Float64(x_46_re - x_46_im) * x_46_re)) - t_0);
	else
		tmp = Float64(Float64(fma(Float64(x_46_re / x_46_im), Float64(x_46_re / x_46_im), -3.0) * Float64(x_46_im * x_46_im)) * x_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]}, 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] - t$95$0), $MachinePrecision], Infinity], N[(N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(N[(x$46$re - x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(x$46$re / x$46$im), $MachinePrecision] * N[(x$46$re / x$46$im), $MachinePrecision] + -3.0), $MachinePrecision] * N[(x$46$im * x$46$im), $MachinePrecision]), $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 95.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. 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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   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.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 28.6

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 42.9

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

   8. Applied rewrites 42.9%

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

   9. Taylor expanded in x.im around inf

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

   11. Applied rewrites 100.0%

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

Alternative 2: 59.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \leq -2 \cdot 10^{-319}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im x.im)) x.re)
       (* (+ (* x.re x.im) (* x.im x.re)) x.im))
      -2e-319)
   (* (* x.re x.im) (* 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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319) {
		tmp = (x_46_re * x_46_im) * (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_46re * x_46im) + (x_46im * x_46re)) * x_46im)) <= (-2d-319)) then
        tmp = (x_46re * x_46im) * (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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319) {
		tmp = (x_46_re * x_46_im) * (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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319:
		tmp = (x_46_re * x_46_im) * (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_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im)) <= -2e-319)
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_im * -3.0));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * 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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319)
		tmp = (x_46_re * x_46_im) * (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$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e-319], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * 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)) < -1.99998e-319

   1. Initial program 94.7%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 50.3%

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

   if -1.99998e-319 < (-.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 83.2%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 62.3%

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

Alternative 3: 59.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \leq -2 \cdot 10^{-319}:\\ \;\;\;\;\left(-3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im x.im)) x.re)
       (* (+ (* x.re x.im) (* x.im x.re)) x.im))
      -2e-319)
   (* (* -3.0 (* x.re x.im)) 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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319) {
		tmp = (-3.0 * (x_46_re * x_46_im)) * 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_46re * x_46im) + (x_46im * x_46re)) * x_46im)) <= (-2d-319)) then
        tmp = ((-3.0d0) * (x_46re * x_46im)) * 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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319) {
		tmp = (-3.0 * (x_46_re * x_46_im)) * 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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319:
		tmp = (-3.0 * (x_46_re * x_46_im)) * 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_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im)) <= -2e-319)
		tmp = Float64(Float64(-3.0 * Float64(x_46_re * x_46_im)) * 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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319)
		tmp = (-3.0 * (x_46_re * x_46_im)) * 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$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e-319], N[(N[(-3.0 * N[(x$46$re * x$46$im), $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)) < -1.99998e-319

   1. Initial program 94.7%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 50.2%

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

   if -1.99998e-319 < (-.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 83.2%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 62.3%

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

Alternative 4: 59.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \leq -2 \cdot 10^{-319}:\\ \;\;\;\;-3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im x.im)) x.re)
       (* (+ (* x.re x.im) (* x.im x.re)) x.im))
      -2e-319)
   (* -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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319) {
		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_46re * x_46im) + (x_46im * x_46re)) * x_46im)) <= (-2d-319)) 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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319) {
		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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319:
		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_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im)) <= -2e-319)
		tmp = Float64(-3.0 * Float64(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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319)
		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$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e-319], N[(-3.0 * N[(N[(x$46$im * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $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)) < -1.99998e-319

   1. Initial program 94.7%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   if -1.99998e-319 < (-.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 83.2%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 62.3%

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

Alternative 5: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \leq -2 \cdot 10^{-319}:\\ \;\;\;\;-3 \cdot \left(\left(x.im \cdot x.im\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im x.im)) x.re)
       (* (+ (* x.re x.im) (* x.im x.re)) x.im))
      -2e-319)
   (* -3.0 (* (* x.im x.im) x.re))
   (* (* 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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319) {
		tmp = -3.0 * ((x_46_im * x_46_im) * x_46_re);
	} 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_46re * x_46im) + (x_46im * x_46re)) * x_46im)) <= (-2d-319)) then
        tmp = (-3.0d0) * ((x_46im * x_46im) * x_46re)
    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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319) {
		tmp = -3.0 * ((x_46_im * x_46_im) * x_46_re);
	} 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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319:
		tmp = -3.0 * ((x_46_im * x_46_im) * x_46_re)
	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_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im)) <= -2e-319)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im * x_46_im) * x_46_re));
	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_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)) <= -2e-319)
		tmp = -3.0 * ((x_46_im * x_46_im) * x_46_re);
	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$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e-319], N[(-3.0 * N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision]), $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)) < -1.99998e-319

   1. Initial program 94.7%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 45.2%

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

   if -1.99998e-319 < (-.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 83.2%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 62.3%

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

Alternative 6: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 4.4 \cdot 10^{+193}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 4.4e+193)
   (-
    (* (+ x.im x.re) (* (- x.re x.im) x.re))
    (* (+ (* x.re x.im) (* 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 <= 4.4e+193) {
		tmp = ((x_46_im + x_46_re) * ((x_46_re - 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_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 <= 4.4d+193) then
        tmp = ((x_46im + x_46re) * ((x_46re - x_46im) * x_46re)) - (((x_46re * x_46im) + (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 <= 4.4e+193) {
		tmp = ((x_46_im + x_46_re) * ((x_46_re - 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_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 4.4e+193:
		tmp = ((x_46_im + x_46_re) * ((x_46_re - 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_re * x_46_re) * x_46_re
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 4.4e+193)
		tmp = Float64(Float64(Float64(x_46_im + x_46_re) * Float64(Float64(x_46_re - 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));
	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 <= 4.4e+193)
		tmp = ((x_46_im + x_46_re) * ((x_46_re - 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_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 4.4e+193], N[(N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(N[(x$46$re - x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision]), $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], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.39999999999999972e193

   1. Initial program 89.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if 4.39999999999999972e193 < x.re

   1. Initial program 61.9%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 5.1

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 81.0

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

   8. Applied rewrites 81.0%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 95.2%

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

Alternative 7: 92.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-3 \cdot x.re\right) \cdot x.im\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 7.6e+153)
   (* (fma -3.0 (* x.im x.im) (* x.re x.re)) x.re)
   (* (* (* -3.0 x.re) x.im) x.im)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 7.6e+153) {
		tmp = fma(-3.0, (x_46_im * x_46_im), (x_46_re * x_46_re)) * x_46_re;
	} else {
		tmp = ((-3.0 * x_46_re) * x_46_im) * x_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 7.6e+153)
		tmp = Float64(fma(-3.0, Float64(x_46_im * x_46_im), Float64(x_46_re * x_46_re)) * x_46_re);
	else
		tmp = Float64(Float64(Float64(-3.0 * x_46_re) * x_46_im) * x_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 7.6e+153], N[(N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(N[(-3.0 * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.59999999999999933e153

   1. Initial program 90.8%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 94.8

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

   8. Applied rewrites 94.8%

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

   if 7.59999999999999933e153 < x.im

   1. Initial program 58.1%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 85.5%

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

   9. Applied rewrites 85.5%

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

Alternative 8: 59.9% 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 87.2%

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

3. Taylor expanded in x.re around 0

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

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

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 58.2

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in x.re around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+r- N/A

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

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

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 90.8

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

8. Applied rewrites 90.8%

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

9. Taylor expanded in x.re around inf

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

11. Applied rewrites 59.0%

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

Alternative 9: 24.3% 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 87.2%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. flip-+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. +-inverses N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-neg-frac2 N/A

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

4. Applied rewrites 56.5%

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

5. Taylor expanded in x.re around 0

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-rgt-identity N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 23.9

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

7. Applied rewrites 23.9%

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

Developer Target 1: 87.2% 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 2024318 
(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)))