math.cube on complex, imaginary part

Percentage Accurate: 82.9% → 99.8%

Time: 7.5s

Alternatives: 6

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re)) * x_46_re)
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im)) <= Inf)
		tmp = Float64(Float64(Float64(Float64(x_46_re - x_46_im) * x_46_im) * Float64(x_46_im + x_46_re)) + t_0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(-x_46_im) / x_46_re), Float64(x_46_im / x_46_re), 3.0) * Float64(x_46_re * x_46_re)) * x_46_im);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 59.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im\\ t_1 := \left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (+ (* x.im x.re) (* x.im x.re)) x.re)
          (* (- (* x.re x.re) (* x.im x.im)) x.im)))
        (t_1 (* (* (- x.im) x.im) x.im)))
   (if (<= t_0 -1e-310)
     t_1
     (if (<= t_0 INFINITY) (* (* (* 3.0 x.im) x.re) x.re) t_1))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im);
	double t_1 = (-x_46_im * x_46_im) * x_46_im;
	double tmp;
	if (t_0 <= -1e-310) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((3.0 * x_46_im) * x_46_re) * x_46_re;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im);
	double t_1 = (-x_46_im * x_46_im) * x_46_im;
	double tmp;
	if (t_0 <= -1e-310) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((3.0 * x_46_im) * x_46_re) * x_46_re;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im)
	t_1 = (-x_46_im * x_46_im) * x_46_im
	tmp = 0
	if t_0 <= -1e-310:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = ((3.0 * x_46_im) * x_46_re) * x_46_re
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im))
	t_1 = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im)
	tmp = 0.0
	if (t_0 <= -1e-310)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(3.0 * x_46_im) * x_46_re) * x_46_re);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im);
	t_1 = (-x_46_im * x_46_im) * x_46_im;
	tmp = 0.0;
	if (t_0 <= -1e-310)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = ((3.0 * x_46_im) * x_46_re) * x_46_re;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-310], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[(N[(3.0 * x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -9.999999999999969e-311 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 77.9%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 56.2

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

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{-{x.im}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.1%

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

   if -9.999999999999969e-311 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.5%

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

   3. Taylor expanded in x.re around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. cube-mult N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      16. associate-*r/ N/A

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

      17. associate-*l* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 63.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im\\ t_1 := \left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (+ (* x.im x.re) (* x.im x.re)) x.re)
          (* (- (* x.re x.re) (* x.im x.im)) x.im)))
        (t_1 (* (* (- x.im) x.im) x.im)))
   (if (<= t_0 -1e-310)
     t_1
     (if (<= t_0 INFINITY) (* (* 3.0 (* x.im x.re)) x.re) t_1))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im);
	double t_1 = (-x_46_im * x_46_im) * x_46_im;
	double tmp;
	if (t_0 <= -1e-310) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (3.0 * (x_46_im * x_46_re)) * x_46_re;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im);
	double t_1 = (-x_46_im * x_46_im) * x_46_im;
	double tmp;
	if (t_0 <= -1e-310) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (3.0 * (x_46_im * x_46_re)) * x_46_re;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im)
	t_1 = (-x_46_im * x_46_im) * x_46_im
	tmp = 0
	if t_0 <= -1e-310:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = (3.0 * (x_46_im * x_46_re)) * x_46_re
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im))
	t_1 = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im)
	tmp = 0.0
	if (t_0 <= -1e-310)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(3.0 * Float64(x_46_im * x_46_re)) * x_46_re);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im);
	t_1 = (-x_46_im * x_46_im) * x_46_im;
	tmp = 0.0;
	if (t_0 <= -1e-310)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = (3.0 * (x_46_im * x_46_re)) * x_46_re;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-310], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[(3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -9.999999999999969e-311 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 77.9%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 56.2

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

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{-{x.im}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.1%

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

   if -9.999999999999969e-311 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.5%

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

   3. Taylor expanded in x.re around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. cube-mult N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      16. associate-*r/ N/A

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

      17. associate-*l* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 58.2%

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

   9. Applied rewrites 63.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 4.7 \cdot 10^{+222}:\\ \;\;\;\;\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 4.7e+222)
   (+
    (* (* (- x.re x.im) x.im) (+ x.im x.re))
    (* (+ (* x.im x.re) (* x.im x.re)) x.re))
   (* (* (- x.im) x.im) x.im)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.7e+222) {
		tmp = (((x_46_re - x_46_im) * x_46_im) * (x_46_im + x_46_re)) + (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re);
	} else {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	}
	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 <= 4.7d+222) then
        tmp = (((x_46re - x_46im) * x_46im) * (x_46im + x_46re)) + (((x_46im * x_46re) + (x_46im * x_46re)) * x_46re)
    else
        tmp = (-x_46im * x_46im) * x_46im
    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 <= 4.7e+222) {
		tmp = (((x_46_re - x_46_im) * x_46_im) * (x_46_im + x_46_re)) + (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re);
	} else {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 4.7e+222:
		tmp = (((x_46_re - x_46_im) * x_46_im) * (x_46_im + x_46_re)) + (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re)
	else:
		tmp = (-x_46_im * x_46_im) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 4.7e+222)
		tmp = Float64(Float64(Float64(Float64(x_46_re - x_46_im) * x_46_im) * Float64(x_46_im + x_46_re)) + Float64(Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re)) * x_46_re));
	else
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 4.7e+222)
		tmp = (((x_46_re - x_46_im) * x_46_im) * (x_46_im + x_46_re)) + (((x_46_im * x_46_re) + (x_46_im * x_46_re)) * x_46_re);
	else
		tmp = (-x_46_im * x_46_im) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 4.7e+222], N[(N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.6999999999999999e222

   1. Initial program 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 96.8

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

   4. Applied rewrites 96.8%

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

   if 4.6999999999999999e222 < x.im

   1. Initial program 57.1%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 95.2

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

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{-{x.im}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.7 \cdot 10^{+222}:\\ \;\;\;\;\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.15 \cdot 10^{-104}:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(-x.im, x.im, \left(3 \cdot x.re\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 2.15e-104)
   (* (* (* 3.0 x.im) x.re) x.re)
   (if (<= x.im 4e+244)
     (* (fma (- x.im) x.im (* (* 3.0 x.re) x.re)) x.im)
     (* (* (- x.im) x.im) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.15e-104) {
		tmp = ((3.0 * x_46_im) * x_46_re) * x_46_re;
	} else if (x_46_im <= 4e+244) {
		tmp = fma(-x_46_im, x_46_im, ((3.0 * x_46_re) * x_46_re)) * x_46_im;
	} else {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.15e-104)
		tmp = Float64(Float64(Float64(3.0 * x_46_im) * x_46_re) * x_46_re);
	elseif (x_46_im <= 4e+244)
		tmp = Float64(fma(Float64(-x_46_im), x_46_im, Float64(Float64(3.0 * x_46_re) * x_46_re)) * x_46_im);
	else
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 2.15e-104], N[(N[(N[(3.0 * x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[x$46$im, 4e+244], N[(N[((-x$46$im) * x$46$im + N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < 2.15000000000000005e-104

   1. Initial program 88.1%

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

   3. Taylor expanded in x.re around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. cube-mult N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      16. associate-*r/ N/A

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

      17. associate-*l* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

   5. Applied rewrites 60.1%

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

   7. Applied rewrites 66.6%

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

   if 2.15000000000000005e-104 < x.im < 4.0000000000000003e244

   1. Initial program 92.1%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 99.8%

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

   if 4.0000000000000003e244 < x.im

   1. Initial program 57.9%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 94.7

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

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{-{x.im}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.15 \cdot 10^{-104}:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(-x.im, x.im, \left(3 \cdot x.re\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return (-x_46_im * 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_46im * x_46im) * x_46im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-pow.f64 62.3

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

5. Applied rewrites 62.3%

   \[\leadsto \color{blue}{-{x.im}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 62.3%

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

Developer Target 1: 92.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024283 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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