math.cube on complex, imaginary part

Percentage Accurate: 82.0% → 97.5%

Time: 7.2s

Alternatives: 13

Speedup: 1.3×

• 45.2% 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: 82.0% 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: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+234}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 1.22 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(x.re, 2 \cdot \left(x.im \cdot x.re\right), \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-{x.im}^{3}\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -3.4e+234)
   (* x.re (* x.im x.im))
   (if (<= x.im 1.22e+207)
     (fma x.re (* 2.0 (* x.im x.re)) (* (+ x.im x.re) (* x.im (- x.re x.im))))
     (- (pow x.im 3.0)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -3.4e+234) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else if (x_46_im <= 1.22e+207) {
		tmp = fma(x_46_re, (2.0 * (x_46_im * x_46_re)), ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im))));
	} else {
		tmp = -pow(x_46_im, 3.0);
	}
	return tmp;
}

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -3.4e+234)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	elseif (x_46_im <= 1.22e+207)
		tmp = fma(x_46_re, Float64(2.0 * Float64(x_46_im * x_46_re)), Float64(Float64(x_46_im + x_46_re) * Float64(x_46_im * Float64(x_46_re - x_46_im))));
	else
		tmp = Float64(-(x_46_im ^ 3.0));
	end
	return tmp
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -3.4e+234], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.22e+207], N[(x$46$re * N[(2.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im + x$46$re), $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 3 regimes

2. if x.im < -3.4e234

   1. Initial program 66.7%

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

      1. +-commutative 66.7%

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

      2. *-commutative 66.7%

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

      3. fma-def 66.7%

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

      4. *-commutative 66.7%

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

      5. count-2 66.7%

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

      6. sqr-neg 66.7%

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

      7. sqr-neg 66.7%

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

      8. sqr-neg 66.7%

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

      9. difference-of-squares 66.7%

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

      10. sub-neg 66.7%

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

      11. sub-neg 66.7%

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

      12. difference-of-squares 66.7%

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

      13. sqr-neg 66.7%

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

      14. difference-of-squares 66.7%

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

      15. +-commutative 66.7%

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

      16. associate-*l* 66.7%

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

      17. +-commutative 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in x.re around inf 6.7%

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

   5. Taylor expanded in x.re around 0 20.0%

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

      1. *-commutative 20.0%

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

      2. unpow2 20.0%

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

   7. Simplified 20.0%

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

   if -3.4e234 < x.im < 1.21999999999999993e207

   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. Step-by-step derivation

      1. +-commutative 89.5%

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

      2. *-commutative 89.5%

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

      3. fma-def 90.0%

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

      4. *-commutative 90.0%

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

      5. count-2 90.0%

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

      6. sqr-neg 90.0%

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

      7. sqr-neg 90.0%

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

      8. sqr-neg 90.0%

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

      9. difference-of-squares 94.6%

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

      10. sub-neg 94.6%

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

      11. sub-neg 94.6%

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

      12. difference-of-squares 90.0%

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

      13. sqr-neg 90.0%

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

      14. difference-of-squares 94.6%

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

      15. +-commutative 94.6%

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

      16. associate-*l* 99.3%

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

      17. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   if 1.21999999999999993e207 < x.im

   1. Initial program 76.0%

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

   2. Taylor expanded in x.re around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+234}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 1.22 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(x.re, 2 \cdot \left(x.im \cdot x.re\right), \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-{x.im}^{3}\\ \end{array} \]

Alternative 2: 91.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq 2.4 \cdot 10^{+112}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re, 2 \cdot \left(x.im \cdot x.re\right), \left(x.im + x.re\right) \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 2.4e+112)
   (- (* x.re (* x.re (* x.im 3.0))) (pow x.im 3.0))
   (fma x.re (* 2.0 (* x.im x.re)) (* (+ x.im x.re) (* x.im x.re)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 2.4e+112) {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - pow(x_46_im, 3.0);
	} else {
		tmp = fma(x_46_re, (2.0 * (x_46_im * x_46_re)), ((x_46_im + x_46_re) * (x_46_im * x_46_re)));
	}
	return tmp;
}

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 2.4e+112)
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0))) - (x_46_im ^ 3.0));
	else
		tmp = fma(x_46_re, Float64(2.0 * Float64(x_46_im * x_46_re)), Float64(Float64(x_46_im + x_46_re) * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 2.4e+112], N[(N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(2.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.4e112

   1. Initial program 90.6%

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

      1. +-commutative 90.6%

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

      2. *-commutative 90.6%

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

      3. distribute-lft-out 90.6%

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

      4. associate-*l* 90.5%

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

      5. *-commutative 90.5%

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

      6. distribute-rgt-out 91.5%

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

      7. associate-+r- 91.5%

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

      8. distribute-lft-out-- 88.9%

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

   3. Simplified 91.8%

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

   if 2.4e112 < x.re

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. *-commutative 60.3%

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

      3. fma-def 63.4%

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

      4. *-commutative 63.4%

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

      5. count-2 63.4%

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

      6. sqr-neg 63.4%

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

      7. sqr-neg 63.4%

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

      8. sqr-neg 63.4%

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

      9. difference-of-squares 79.1%

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

      10. sub-neg 79.1%

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

      11. sub-neg 79.1%

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

      12. difference-of-squares 63.4%

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

      13. sqr-neg 63.4%

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

      14. difference-of-squares 79.1%

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

      15. +-commutative 79.1%

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

      16. associate-*l* 93.6%

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

      17. +-commutative 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in x.re around inf 93.6%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.4 \cdot 10^{+112}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re, 2 \cdot \left(x.im \cdot x.re\right), \left(x.im + x.re\right) \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 3: 90.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{if}\;x.re \leq 8.6 \cdot 10^{+61}:\\ \;\;\;\;t_0 - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* x.re (* x.im 3.0)))))
   (if (<= x.re 8.6e+61) (- t_0 (pow x.im 3.0)) t_0)))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * (x_46_im * 3.0));
	double tmp;
	if (x_46_re <= 8.6e+61) {
		tmp = t_0 - pow(x_46_im, 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re * (x_46re * (x_46im * 3.0d0))
    if (x_46re <= 8.6d+61) then
        tmp = t_0 - (x_46im ** 3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * (x_46_im * 3.0));
	double tmp;
	if (x_46_re <= 8.6e+61) {
		tmp = t_0 - Math.pow(x_46_im, 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	t_0 = x_46_re * (x_46_re * (x_46_im * 3.0))
	tmp = 0
	if x_46_re <= 8.6e+61:
		tmp = t_0 - math.pow(x_46_im, 3.0)
	else:
		tmp = t_0
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)))
	tmp = 0.0
	if (x_46_re <= 8.6e+61)
		tmp = Float64(t_0 - (x_46_im ^ 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * (x_46_re * (x_46_im * 3.0));
	tmp = 0.0;
	if (x_46_re <= 8.6e+61)
		tmp = t_0 - (x_46_im ^ 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, 8.6e+61], N[(t$95$0 - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if x.re < 8.6000000000000003e61

   1. Initial program 90.7%

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

      1. +-commutative 90.7%

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

      2. *-commutative 90.7%

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

      3. distribute-lft-out 90.7%

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

      4. associate-*l* 90.7%

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

      5. *-commutative 90.7%

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

      6. distribute-rgt-out 91.2%

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

      7. associate-+r- 91.2%

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

      8. distribute-lft-out-- 88.9%

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

   3. Simplified 92.0%

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

   if 8.6000000000000003e61 < x.re

   1. Initial program 65.6%

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

   2. Taylor expanded in x.re around inf 78.2%

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

      1. unpow2 78.2%

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

      2. distribute-rgt1-in 78.2%

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

      3. metadata-eval 78.2%

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

      4. *-commutative 78.2%

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

      5. associate-*r* 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 8.6 \cdot 10^{+61}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 4: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.8 \cdot 10^{+78}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 1.8e+78)
   (+
    (* x.im (- (* x.re x.re) (* x.im x.im)))
    (* x.re (+ (* x.im x.re) (* x.im x.re))))
   (* x.re (* x.re (* x.im 3.0)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.8e+78) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 1.8d+78) then
        tmp = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46re * ((x_46im * x_46re) + (x_46im * x_46re)))
    else
        tmp = x_46re * (x_46re * (x_46im * 3.0d0))
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.8e+78) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 1.8e+78:
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)))
	else:
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0))
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 1.8e+78)
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re))));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 1.8e+78)
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	else
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 1.8e+78], N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.8000000000000001e78

   1. Initial program 90.8%

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

   if 1.8000000000000001e78 < x.re

   1. Initial program 64.7%

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

   2. Taylor expanded in x.re around inf 77.6%

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

      1. unpow2 77.6%

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

      2. distribute-rgt1-in 77.6%

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

      3. metadata-eval 77.6%

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

      4. *-commutative 77.6%

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

      5. associate-*r* 89.7%

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

   4. Simplified 89.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.8 \cdot 10^{+78}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 5: 83.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq 26000:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re + x.re\right)\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 26000.0)
   (- (* x.re (* x.im (+ x.re x.re))) (* x.im (* x.im x.im)))
   (* x.re (* x.re (* x.im 3.0)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 26000.0) {
		tmp = (x_46_re * (x_46_im * (x_46_re + x_46_re))) - (x_46_im * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
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 <= 26000.0d0) then
        tmp = (x_46re * (x_46im * (x_46re + x_46re))) - (x_46im * (x_46im * x_46im))
    else
        tmp = x_46re * (x_46re * (x_46im * 3.0d0))
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 26000.0) {
		tmp = (x_46_re * (x_46_im * (x_46_re + x_46_re))) - (x_46_im * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 26000.0:
		tmp = (x_46_re * (x_46_im * (x_46_re + x_46_re))) - (x_46_im * (x_46_im * x_46_im))
	else:
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0))
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 26000.0)
		tmp = Float64(Float64(x_46_re * Float64(x_46_im * Float64(x_46_re + x_46_re))) - Float64(x_46_im * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 26000.0)
		tmp = (x_46_re * (x_46_im * (x_46_re + x_46_re))) - (x_46_im * (x_46_im * x_46_im));
	else
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 26000.0], N[(N[(x$46$re * N[(x$46$im * N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 26000

   1. Initial program 90.4%

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

   2. Taylor expanded in x.re around 0 75.1%

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

      1. mul-1-neg 75.1%

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

      2. unpow2 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

   4. Simplified 75.1%

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

   5. Taylor expanded in x.re around 0 75.1%

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

      1. count-2 75.1%

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

      2. distribute-lft-out 75.1%

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

   7. Simplified 75.1%

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

   if 26000 < x.re

   1. Initial program 71.3%

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

   2. Taylor expanded in x.re around inf 77.6%

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

      1. unpow2 77.6%

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

      2. distribute-rgt1-in 77.6%

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

      3. metadata-eval 77.6%

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

      4. *-commutative 77.6%

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

      5. associate-*r* 87.3%

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

   4. Simplified 87.3%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 26000:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re + x.re\right)\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 6: 58.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4.2 \cdot 10^{+162}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -4.2e+162)
   (* x.re (* x.im x.im))
   (* x.im (* x.re (* x.re 3.0)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -4.2e+162) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0));
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
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 <= (-4.2d+162)) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = x_46im * (x_46re * (x_46re * 3.0d0))
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -4.2e+162) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0));
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -4.2e+162:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0))
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -4.2e+162)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_re * 3.0)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -4.2e+162)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -4.2e+162], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.2000000000000001e162

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

      2. *-commutative 61.8%

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

      3. fma-def 64.7%

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

      4. *-commutative 64.7%

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

      5. count-2 64.7%

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

      6. sqr-neg 64.7%

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

      7. sqr-neg 64.7%

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

      8. sqr-neg 64.7%

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

      9. difference-of-squares 82.4%

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

      10. sub-neg 82.4%

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

      11. sub-neg 82.4%

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

      12. difference-of-squares 64.7%

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

      13. sqr-neg 64.7%

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

      14. difference-of-squares 82.4%

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

      15. +-commutative 82.4%

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

      16. associate-*l* 82.4%

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

      17. +-commutative 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in x.re around inf 26.5%

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

   5. Taylor expanded in x.re around 0 23.5%

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

      1. *-commutative 23.5%

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

      2. unpow2 23.5%

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

   7. Simplified 23.5%

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

   if -4.2000000000000001e162 < x.im

   1. Initial program 90.6%

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

      1. +-commutative 90.6%

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

      2. *-commutative 90.6%

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

      3. fma-def 90.7%

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

      4. *-commutative 90.7%

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

      5. count-2 90.7%

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

      6. sqr-neg 90.7%

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

      7. sqr-neg 90.7%

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

      8. sqr-neg 90.7%

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

      9. difference-of-squares 92.5%

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

      10. sub-neg 92.5%

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

      11. sub-neg 92.5%

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

      12. difference-of-squares 90.7%

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

      13. sqr-neg 90.7%

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

      14. difference-of-squares 92.5%

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

      15. +-commutative 92.5%

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

      16. associate-*l* 97.1%

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

      17. +-commutative 97.1%

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

   3. Simplified 97.1%

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

      1. add-cbrt-cube_binary64 75.9%

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

   5. Applied rewrite-once 75.9%

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

   6. Taylor expanded in x.re around inf 56.5%

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

      1. *-commutative 56.5%

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

      2. distribute-rgt1-in 56.5%

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

      3. metadata-eval 56.5%

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

      4. *-commutative 56.5%

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

      5. associate-*r* 56.1%

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

      6. unpow2 56.1%

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

      7. *-commutative 56.1%

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

      8. associate-*l* 56.1%

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

   8. Simplified 56.1%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.2 \cdot 10^{+162}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 7: 58.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4.2 \cdot 10^{+162}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -4.2e+162)
   (* x.re (* x.im x.im))
   (* x.im (* 3.0 (* x.re x.re)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -4.2e+162) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
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 <= (-4.2d+162)) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = x_46im * (3.0d0 * (x_46re * x_46re))
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -4.2e+162) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -4.2e+162:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re))
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -4.2e+162)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(x_46_im * Float64(3.0 * Float64(x_46_re * x_46_re)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -4.2e+162)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -4.2e+162], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.2000000000000001e162

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

      2. *-commutative 61.8%

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

      3. fma-def 64.7%

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

      4. *-commutative 64.7%

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

      5. count-2 64.7%

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

      6. sqr-neg 64.7%

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

      7. sqr-neg 64.7%

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

      8. sqr-neg 64.7%

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

      9. difference-of-squares 82.4%

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

      10. sub-neg 82.4%

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

      11. sub-neg 82.4%

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

      12. difference-of-squares 64.7%

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

      13. sqr-neg 64.7%

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

      14. difference-of-squares 82.4%

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

      15. +-commutative 82.4%

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

      16. associate-*l* 82.4%

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

      17. +-commutative 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in x.re around inf 26.5%

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

   5. Taylor expanded in x.re around 0 23.5%

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

      1. *-commutative 23.5%

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

      2. unpow2 23.5%

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

   7. Simplified 23.5%

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

   if -4.2000000000000001e162 < x.im

   1. Initial program 90.6%

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

   2. Taylor expanded in x.re around inf 56.5%

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

      1. distribute-rgt1-in 56.5%

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

      2. metadata-eval 56.5%

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

      3. *-commutative 56.5%

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

      4. associate-*r* 56.5%

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

      5. *-commutative 56.5%

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

      6. associate-*l* 56.1%

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

      7. unpow2 56.1%

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

   4. Simplified 56.1%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.2 \cdot 10^{+162}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]

Alternative 8: 64.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4.2 \cdot 10^{+162}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -4.2e+162)
   (* x.re (* x.im x.im))
   (* x.re (* 3.0 (* x.im x.re)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -4.2e+162) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re));
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
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 <= (-4.2d+162)) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = x_46re * (3.0d0 * (x_46im * x_46re))
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -4.2e+162) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -4.2e+162:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re))
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -4.2e+162)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(x_46_re * Float64(3.0 * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -4.2e+162)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -4.2e+162], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.2000000000000001e162

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

      2. *-commutative 61.8%

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

      3. fma-def 64.7%

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

      4. *-commutative 64.7%

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

      5. count-2 64.7%

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

      6. sqr-neg 64.7%

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

      7. sqr-neg 64.7%

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

      8. sqr-neg 64.7%

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

      9. difference-of-squares 82.4%

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

      10. sub-neg 82.4%

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

      11. sub-neg 82.4%

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

      12. difference-of-squares 64.7%

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

      13. sqr-neg 64.7%

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

      14. difference-of-squares 82.4%

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

      15. +-commutative 82.4%

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

      16. associate-*l* 82.4%

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

      17. +-commutative 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in x.re around inf 26.5%

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

   5. Taylor expanded in x.re around 0 23.5%

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

      1. *-commutative 23.5%

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

      2. unpow2 23.5%

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

   7. Simplified 23.5%

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

   if -4.2000000000000001e162 < x.im

   1. Initial program 90.6%

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

   2. Taylor expanded in x.re around inf 56.5%

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

      1. unpow2 56.5%

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

      2. distribute-rgt1-in 56.5%

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

      3. metadata-eval 56.5%

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

      4. *-commutative 56.5%

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

      5. associate-*r* 61.1%

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

   4. Simplified 61.1%

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

   5. Taylor expanded in x.re around 0 61.1%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.2 \cdot 10^{+162}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 9: 64.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+162}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -5e+162) (* x.re (* x.im x.im)) (* x.re (* x.re (* x.im 3.0)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -5e+162) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
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 <= (-5d+162)) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = x_46re * (x_46re * (x_46im * 3.0d0))
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -5e+162) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -5e+162:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0))
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -5e+162)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -5e+162)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -5e+162], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.9999999999999997e162

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

      2. *-commutative 61.8%

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

      3. fma-def 64.7%

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

      4. *-commutative 64.7%

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

      5. count-2 64.7%

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

      6. sqr-neg 64.7%

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

      7. sqr-neg 64.7%

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

      8. sqr-neg 64.7%

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

      9. difference-of-squares 82.4%

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

      10. sub-neg 82.4%

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

      11. sub-neg 82.4%

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

      12. difference-of-squares 64.7%

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

      13. sqr-neg 64.7%

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

      14. difference-of-squares 82.4%

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

      15. +-commutative 82.4%

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

      16. associate-*l* 82.4%

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

      17. +-commutative 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in x.re around inf 26.5%

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

   5. Taylor expanded in x.re around 0 23.5%

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

      1. *-commutative 23.5%

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

      2. unpow2 23.5%

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

   7. Simplified 23.5%

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

   if -4.9999999999999997e162 < x.im

   1. Initial program 90.6%

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

   2. Taylor expanded in x.re around inf 56.5%

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

      1. unpow2 56.5%

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

      2. distribute-rgt1-in 56.5%

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

      3. metadata-eval 56.5%

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

      4. *-commutative 56.5%

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

      5. associate-*r* 61.1%

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

   4. Simplified 61.1%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+162}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 10: 27.7% accurate, 3.8× speedup

Math

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

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 (* x.im (* x.im x.re)))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_im * x_46_re);
}

Fortran

NOTE: x.re should be positive before calling this function
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

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_im * x_46_re);
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	return x_46_im * (x_46_im * x_46_re)

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	return Float64(x_46_im * Float64(x_46_im * x_46_re))
end

MATLAB

x.re = abs(x.re)
function tmp = code(x_46_re, x_46_im)
	tmp = x_46_im * (x_46_im * x_46_re);
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $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. Step-by-step derivation

   1. +-commutative 86.8%

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

   2. *-commutative 86.8%

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

   3. fma-def 87.2%

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

   4. *-commutative 87.2%

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

   5. count-2 87.2%

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

   6. sqr-neg 87.2%

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

   7. sqr-neg 87.2%

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

   8. sqr-neg 87.2%

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

   9. difference-of-squares 91.1%

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

   10. sub-neg 91.1%

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

   11. sub-neg 91.1%

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

   12. difference-of-squares 87.2%

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

   13. sqr-neg 87.2%

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

   14. difference-of-squares 91.1%

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

   15. +-commutative 91.1%

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

   16. associate-*l* 95.1%

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

   17. +-commutative 95.1%

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

3. Simplified 95.1%

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

4. Taylor expanded in x.re around inf 59.6%

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

5. Taylor expanded in x.re around 0 27.4%

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

   1. unpow2 27.4%

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

   2. associate-*l* 25.6%

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

7. Simplified 25.6%

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

8. Final simplification 25.6%

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

Alternative 11: 30.6% accurate, 3.8× speedup

Math

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

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 (* x.re (* x.im x.im)))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_im * x_46_im);
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = x_46re * (x_46im * x_46im)
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_im * x_46_im);
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	return x_46_re * (x_46_im * x_46_im)

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	return Float64(x_46_re * Float64(x_46_im * x_46_im))
end

MATLAB

x.re = abs(x.re)
function tmp = code(x_46_re, x_46_im)
	tmp = x_46_re * (x_46_im * x_46_im);
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $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. Step-by-step derivation

   1. +-commutative 86.8%

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

   2. *-commutative 86.8%

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

   3. fma-def 87.2%

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

   4. *-commutative 87.2%

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

   5. count-2 87.2%

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

   6. sqr-neg 87.2%

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

   7. sqr-neg 87.2%

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

   8. sqr-neg 87.2%

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

   9. difference-of-squares 91.1%

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

   10. sub-neg 91.1%

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

   11. sub-neg 91.1%

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

   12. difference-of-squares 87.2%

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

   13. sqr-neg 87.2%

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

   14. difference-of-squares 91.1%

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

   15. +-commutative 91.1%

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

   16. associate-*l* 95.1%

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

   17. +-commutative 95.1%

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

3. Simplified 95.1%

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

4. Taylor expanded in x.re around inf 59.6%

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

5. Taylor expanded in x.re around 0 27.4%

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

   1. *-commutative 27.4%

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

   2. unpow2 27.4%

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

7. Simplified 27.4%

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

8. Final simplification 27.4%

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

Alternative 12: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ -1 \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 -1.0)

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	return -1.0;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -1.0d0
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	return -1.0;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	return -1.0

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	return -1.0
end

MATLAB

x.re = abs(x.re)
function tmp = code(x_46_re, x_46_im)
	tmp = -1.0;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := -1.0

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. Taylor expanded in x.re around 0 67.1%

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

   1. mul-1-neg 67.1%

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

   2. unpow2 67.1%

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

   3. distribute-rgt-neg-in 67.1%

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

4. Simplified 67.1%

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

5. Taylor expanded in x.re around 0 67.1%

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

   1. count-2 67.1%

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

   2. distribute-lft-out 67.1%

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

7. Simplified 67.1%

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

9. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-1} \]

10. Final simplification 2.9%

    \[\leadsto -1 \]

Alternative 13: 15.1% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ 0 \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 0.0)

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	return 0.0;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = 0.0d0
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	return 0.0;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	return 0.0

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	return 0.0
end

MATLAB

x.re = abs(x.re)
function tmp = code(x_46_re, x_46_im)
	tmp = 0.0;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := 0.0

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. Taylor expanded in x.re around 0 67.1%

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

   1. mul-1-neg 67.1%

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

   2. unpow2 67.1%

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

   3. distribute-rgt-neg-in 67.1%

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

4. Simplified 67.1%

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

5. Taylor expanded in x.re around 0 67.1%

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

   1. count-2 67.1%

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

   2. distribute-lft-out 67.1%

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

7. Simplified 67.1%

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

9. Applied egg-rr 15.2%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 15.2%

    \[\leadsto 0 \]

Developer target: 91.0% 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 2023297 
(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)))