math.cube on complex, imaginary part

Percentage Accurate: 83.0% → 96.7%

Time: 6.4s

Alternatives: 4

Speedup: 1.0×

• 44.3% 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: 83.0% 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: 96.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))
	tmp = 0
	if ((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0) <= math.inf:
		tmp = t_0 + ((x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im)))
	else:
		tmp = -math.pow(x_46_im, 3.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	tmp = 0.0;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0) <= Inf)
		tmp = t_0 + ((x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im)));
	else
		tmp = -(x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.1%

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

      1. difference-of-squares 93.1%

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

      2. *-commutative 93.1%

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

   4. Applied egg-rr 93.1%

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

      1. *-commutative 93.1%

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

      2. distribute-rgt-in 90.3%

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

      3. distribute-lft-in 84.4%

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

   6. Applied egg-rr 84.4%

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

   8. Simplified 90.3%

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

      1. +-commutative 90.3%

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

      2. distribute-rgt-in 84.4%

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

      3. *-commutative 84.4%

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

      4. associate-*l* 84.4%

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

      5. pow2 84.4%

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

      6. *-commutative 84.4%

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

      7. associate-*l* 91.2%

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

      8. *-commutative 91.2%

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

   10. Applied egg-rr 91.2%

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

   12. Simplified 99.8%

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

   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%

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

   3. Simplified 0.0%

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

      1. *-commutative 0.0%

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

      2. fma-neg 19.4%

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

      3. associate-*r* 19.4%

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

      4. *-commutative 19.4%

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

      5. associate-*l* 19.4%

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

   6. Applied egg-rr 19.4%

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

   7. Taylor expanded in x.re around 0 80.6%

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

      1. mul-1-neg 80.6%

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

   9. Simplified 80.6%

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

4. Final simplification 97.1%

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

Alternative 2: 86.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+
  (* x.re (+ (* x.re x.im) (* x.re x.im)))
  (* 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_re * x_46_im) + (x_46_re * x_46_im))) + (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_46re * x_46im) + (x_46re * x_46im))) + (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_re * x_46_im) + (x_46_re * x_46_im))) + (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_re * x_46_im) + (x_46_re * x_46_im))) + (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(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))) + Float64(x_46_im * Float64(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_re * x_46_im) + (x_46_re * x_46_im))) + (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[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. difference-of-squares 82.7%

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

   2. *-commutative 82.7%

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

4. Applied egg-rr 82.7%

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

5. Final simplification 82.7%

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

Alternative 3: 91.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+
  (* x.re (+ (* x.re x.im) (* x.re x.im)))
  (* (- 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_re * x_46_im))) + ((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_46re * x_46im))) + ((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_re * x_46_im))) + ((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_re * x_46_im))) + ((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(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))) + Float64(Float64(x_46_re - x_46_im) * Float64(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_re * x_46_im) + (x_46_re * x_46_im))) + ((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[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. difference-of-squares 82.7%

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

   2. *-commutative 82.7%

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

4. Applied egg-rr 82.7%

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

   1. *-commutative 82.7%

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

   2. distribute-rgt-in 78.8%

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

   3. distribute-lft-in 73.7%

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

6. Applied egg-rr 73.7%

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

8. Simplified 78.8%

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

   1. +-commutative 78.8%

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

   2. distribute-rgt-in 73.7%

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

   3. *-commutative 73.7%

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

   4. associate-*l* 73.7%

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

   5. pow2 73.7%

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

   6. *-commutative 73.7%

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

   7. associate-*l* 79.5%

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

   8. *-commutative 79.5%

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

10. Applied egg-rr 79.5%

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

12. Simplified 88.5%

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

13. Final simplification 88.5%

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

Alternative 4: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x_46_re, x_46_im):
	return -3.0

Julia

function code(x_46_re, x_46_im)
	return -3.0
end

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := -3.0

Derivation

1. Initial program 80.0%

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

3. Simplified 83.9%

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

5. Taylor expanded in x.re around 0 61.4%

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]

7. Simplified 2.7%

   \[\leadsto \color{blue}{-3} \]

8. Final simplification 2.7%

   \[\leadsto -3 \]
9. Add Preprocessing

Developer target: 91.5% 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 2024031 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 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)))