math.cube on complex, real part

Percentage Accurate: 81.7% → 96.1%

Time: 8.0s

Alternatives: 9

Speedup: 0.9×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.2 \cdot 10^{+147}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(-3, x.im \cdot x.im, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.2e+147)
   (* x.re (fma -3.0 (* x.im x.im) (* x.re x.re)))
   (* -3.0 (* x.im (* x.im x.re)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.2e+147) {
		tmp = x_46_re * fma(-3.0, (x_46_im * x_46_im), (x_46_re * x_46_re));
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	return tmp;
}

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.2e+147)
		tmp = Float64(x_46_re * fma(-3.0, Float64(x_46_im * x_46_im), Float64(x_46_re * x_46_re)));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 1.2e+147], N[(x$46$re * N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.20000000000000001e147

   1. Initial program 86.6%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in x.re around 0 85.8%

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

      1. associate-*r* 85.8%

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

      2. *-commutative 85.8%

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

      3. metadata-eval 85.8%

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

      4. distribute-rgt-out-- 85.8%

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

      5. *-commutative 85.8%

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

      6. fma-def 90.9%

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

      7. distribute-rgt-out-- 90.9%

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

      8. metadata-eval 90.9%

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

      9. unpow2 90.9%

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

      10. associate-*r* 90.8%

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

      11. fma-def 85.8%

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

      12. cube-mult 85.7%

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

      13. unpow2 85.7%

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

      14. distribute-lft-out 91.6%

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

      15. associate-*r* 91.6%

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

      16. unpow2 91.6%

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

      17. *-commutative 91.6%

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

      18. fma-def 91.7%

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

      19. unpow2 91.7%

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

      20. unpow2 91.7%

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

   6. Simplified 91.7%

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

   if 1.20000000000000001e147 < x.im

   1. Initial program 53.8%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in x.re around 0 51.2%

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

      1. *-commutative 51.2%

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

      2. associate-*r* 51.2%

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

      3. *-commutative 51.2%

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

      4. metadata-eval 51.2%

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

      5. distribute-rgt-out-- 51.2%

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

      6. unpow2 51.2%

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

      7. associate-*r* 81.8%

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

      8. distribute-rgt-out-- 81.8%

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

      9. metadata-eval 81.8%

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

      10. *-commutative 81.8%

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

   6. Simplified 81.8%

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

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

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

   8. Taylor expanded in x.re around 0 61.5%

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

      1. *-commutative 61.5%

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

      2. unpow2 61.5%

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

      3. associate-*r* 92.1%

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

      4. *-commutative 92.1%

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

   10. Simplified 92.1%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.2 \cdot 10^{+147}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(-3, x.im \cdot x.im, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 2: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re\right)\\ t_1 := t_0 - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{if}\;x.re \leq -2.55 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -1.45 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq 2.1 \cdot 10^{-75}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* x.re x.re)))
        (t_1 (- t_0 (* x.im (+ (* x.im x.re) (* x.im x.re))))))
   (if (<= x.re -2.55e+120)
     t_0
     (if (<= x.re -1.45e-98)
       t_1
       (if (<= x.re 2.1e-75)
         (* x.im (* x.im (* x.re -3.0)))
         (if (<= x.re 2e+100) t_1 t_0))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * x_46_re);
	double t_1 = t_0 - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double tmp;
	if (x_46_re <= -2.55e+120) {
		tmp = t_0;
	} else if (x_46_re <= -1.45e-98) {
		tmp = t_1;
	} else if (x_46_re <= 2.1e-75) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else if (x_46_re <= 2e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x.im 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) :: t_1
    real(8) :: tmp
    t_0 = x_46re * (x_46re * x_46re)
    t_1 = t_0 - (x_46im * ((x_46im * x_46re) + (x_46im * x_46re)))
    if (x_46re <= (-2.55d+120)) then
        tmp = t_0
    else if (x_46re <= (-1.45d-98)) then
        tmp = t_1
    else if (x_46re <= 2.1d-75) then
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    else if (x_46re <= 2d+100) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * x_46_re);
	double t_1 = t_0 - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double tmp;
	if (x_46_re <= -2.55e+120) {
		tmp = t_0;
	} else if (x_46_re <= -1.45e-98) {
		tmp = t_1;
	} else if (x_46_re <= 2.1e-75) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else if (x_46_re <= 2e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	t_0 = x_46_re * (x_46_re * x_46_re)
	t_1 = t_0 - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)))
	tmp = 0
	if x_46_re <= -2.55e+120:
		tmp = t_0
	elif x_46_re <= -1.45e-98:
		tmp = t_1
	elif x_46_re <= 2.1e-75:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	elif x_46_re <= 2e+100:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(x_46_re * x_46_re))
	t_1 = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re))))
	tmp = 0.0
	if (x_46_re <= -2.55e+120)
		tmp = t_0;
	elseif (x_46_re <= -1.45e-98)
		tmp = t_1;
	elseif (x_46_re <= 2.1e-75)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	elseif (x_46_re <= 2e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * (x_46_re * x_46_re);
	t_1 = t_0 - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	tmp = 0.0;
	if (x_46_re <= -2.55e+120)
		tmp = t_0;
	elseif (x_46_re <= -1.45e-98)
		tmp = t_1;
	elseif (x_46_re <= 2.1e-75)
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	elseif (x_46_re <= 2e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im 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 * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(x$46$im * N[(N[(x$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -2.55e+120], t$95$0, If[LessEqual[x$46$re, -1.45e-98], t$95$1, If[LessEqual[x$46$re, 2.1e-75], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2e+100], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.55000000000000014e120 or 2.00000000000000003e100 < x.re

   1. Initial program 60.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in x.re around 0 56.8%

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

      1. associate-*r* 56.8%

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

      2. *-commutative 56.8%

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

      3. metadata-eval 56.8%

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

      4. distribute-rgt-out-- 56.8%

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

      5. *-commutative 56.8%

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

      6. fma-def 71.6%

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

      7. distribute-rgt-out-- 71.6%

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

      8. metadata-eval 71.6%

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

      9. unpow2 71.6%

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

      10. associate-*r* 71.6%

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

      11. fma-def 56.8%

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

      12. cube-mult 56.8%

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

      13. unpow2 56.8%

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

      14. distribute-lft-out 75.7%

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

      15. associate-*r* 75.7%

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

      16. unpow2 75.7%

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

      17. *-commutative 75.7%

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

      18. fma-def 75.7%

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

      19. unpow2 75.7%

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

      20. unpow2 75.7%

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

   6. Simplified 75.7%

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

   7. Taylor expanded in x.im around 0 85.1%

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

      1. unpow2 85.1%

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

   9. Simplified 85.1%

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

   if -2.55000000000000014e120 < x.re < -1.45e-98 or 2.1000000000000001e-75 < x.re < 2.00000000000000003e100

   1. Initial program 95.6%

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

   2. Taylor expanded in x.re around inf 87.0%

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

      1. unpow2 87.0%

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

   4. Simplified 87.0%

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

   if -1.45e-98 < x.re < 2.1000000000000001e-75

   1. Initial program 84.9%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in x.re around 0 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*r* 84.8%

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

      3. *-commutative 84.8%

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

      4. metadata-eval 84.8%

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

      5. distribute-rgt-out-- 84.8%

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

      6. unpow2 84.8%

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

      7. associate-*r* 99.7%

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

      8. distribute-rgt-out-- 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x.re around 0 83.8%

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

      1. *-commutative 83.8%

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

      2. associate-*r* 83.9%

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

      3. *-commutative 83.9%

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

      4. unpow2 83.9%

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

      5. associate-*r* 98.7%

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

   9. Simplified 98.7%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.55 \cdot 10^{+120}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq -1.45 \cdot 10^{-98}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq 2.1 \cdot 10^{-75}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+100}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 3: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re\right)\\ t_1 := x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ t_2 := t_0 - t_1\\ \mathbf{if}\;x.re \leq -2.55 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq 6.2 \cdot 10^{-74}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.re\right)\right) - t_1\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* x.re x.re)))
        (t_1 (* x.im (+ (* x.im x.re) (* x.im x.re))))
        (t_2 (- t_0 t_1)))
   (if (<= x.re -2.55e+120)
     t_0
     (if (<= x.re -2.6e-101)
       t_2
       (if (<= x.re 6.2e-74)
         (- (* x.im (* x.im (- x.re))) t_1)
         (if (<= x.re 2e+100) t_2 t_0))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * x_46_re);
	double t_1 = x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re));
	double t_2 = t_0 - t_1;
	double tmp;
	if (x_46_re <= -2.55e+120) {
		tmp = t_0;
	} else if (x_46_re <= -2.6e-101) {
		tmp = t_2;
	} else if (x_46_re <= 6.2e-74) {
		tmp = (x_46_im * (x_46_im * -x_46_re)) - t_1;
	} else if (x_46_re <= 2e+100) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x.im 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x_46re * (x_46re * x_46re)
    t_1 = x_46im * ((x_46im * x_46re) + (x_46im * x_46re))
    t_2 = t_0 - t_1
    if (x_46re <= (-2.55d+120)) then
        tmp = t_0
    else if (x_46re <= (-2.6d-101)) then
        tmp = t_2
    else if (x_46re <= 6.2d-74) then
        tmp = (x_46im * (x_46im * -x_46re)) - t_1
    else if (x_46re <= 2d+100) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * x_46_re);
	double t_1 = x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re));
	double t_2 = t_0 - t_1;
	double tmp;
	if (x_46_re <= -2.55e+120) {
		tmp = t_0;
	} else if (x_46_re <= -2.6e-101) {
		tmp = t_2;
	} else if (x_46_re <= 6.2e-74) {
		tmp = (x_46_im * (x_46_im * -x_46_re)) - t_1;
	} else if (x_46_re <= 2e+100) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	t_0 = x_46_re * (x_46_re * x_46_re)
	t_1 = x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re))
	t_2 = t_0 - t_1
	tmp = 0
	if x_46_re <= -2.55e+120:
		tmp = t_0
	elif x_46_re <= -2.6e-101:
		tmp = t_2
	elif x_46_re <= 6.2e-74:
		tmp = (x_46_im * (x_46_im * -x_46_re)) - t_1
	elif x_46_re <= 2e+100:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(x_46_re * x_46_re))
	t_1 = Float64(x_46_im * Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re)))
	t_2 = Float64(t_0 - t_1)
	tmp = 0.0
	if (x_46_re <= -2.55e+120)
		tmp = t_0;
	elseif (x_46_re <= -2.6e-101)
		tmp = t_2;
	elseif (x_46_re <= 6.2e-74)
		tmp = Float64(Float64(x_46_im * Float64(x_46_im * Float64(-x_46_re))) - t_1);
	elseif (x_46_re <= 2e+100)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * (x_46_re * x_46_re);
	t_1 = x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re));
	t_2 = t_0 - t_1;
	tmp = 0.0;
	if (x_46_re <= -2.55e+120)
		tmp = t_0;
	elseif (x_46_re <= -2.6e-101)
		tmp = t_2;
	elseif (x_46_re <= 6.2e-74)
		tmp = (x_46_im * (x_46_im * -x_46_re)) - t_1;
	elseif (x_46_re <= 2e+100)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im 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 * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(N[(x$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, If[LessEqual[x$46$re, -2.55e+120], t$95$0, If[LessEqual[x$46$re, -2.6e-101], t$95$2, If[LessEqual[x$46$re, 6.2e-74], N[(N[(x$46$im * N[(x$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[x$46$re, 2e+100], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.55000000000000014e120 or 2.00000000000000003e100 < x.re

   1. Initial program 60.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in x.re around 0 56.8%

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

      1. associate-*r* 56.8%

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

      2. *-commutative 56.8%

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

      3. metadata-eval 56.8%

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

      4. distribute-rgt-out-- 56.8%

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

      5. *-commutative 56.8%

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

      6. fma-def 71.6%

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

      7. distribute-rgt-out-- 71.6%

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

      8. metadata-eval 71.6%

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

      9. unpow2 71.6%

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

      10. associate-*r* 71.6%

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

      11. fma-def 56.8%

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

      12. cube-mult 56.8%

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

      13. unpow2 56.8%

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

      14. distribute-lft-out 75.7%

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

      15. associate-*r* 75.7%

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

      16. unpow2 75.7%

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

      17. *-commutative 75.7%

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

      18. fma-def 75.7%

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

      19. unpow2 75.7%

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

      20. unpow2 75.7%

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

   6. Simplified 75.7%

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

   7. Taylor expanded in x.im around 0 85.1%

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

      1. unpow2 85.1%

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

   9. Simplified 85.1%

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

   if -2.55000000000000014e120 < x.re < -2.6000000000000001e-101 or 6.2000000000000003e-74 < x.re < 2.00000000000000003e100

   1. Initial program 95.6%

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

   2. Taylor expanded in x.re around inf 87.0%

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

      1. unpow2 87.0%

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

   4. Simplified 87.0%

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

   if -2.6000000000000001e-101 < x.re < 6.2000000000000003e-74

   1. Initial program 84.9%

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

   2. Taylor expanded in x.re around 0 83.9%

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

      1. mul-1-neg 83.9%

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

      2. unpow2 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

   4. Simplified 83.9%

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

   5. Taylor expanded in x.im around 0 83.9%

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

      1. associate-*r* 83.9%

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

      2. mul-1-neg 83.9%

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

      3. unpow2 83.9%

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

      4. distribute-rgt-neg-out 83.9%

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

      5. associate-*l* 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.55 \cdot 10^{+120}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq 6.2 \cdot 10^{-74}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.re\right)\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+100}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 4: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.35 \cdot 10^{+126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 4.7 \cdot 10^{+142}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 2.35e+126)
   (-
    (* x.re (- (* x.re x.re) (* x.im x.im)))
    (* x.im (+ (* x.im x.re) (* x.im x.re))))
   (if (<= x.im 4.7e+142)
     (* x.re (* x.re x.re))
     (* -3.0 (* x.im (* x.im x.re))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.35e+126) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	} else if (x_46_im <= 4.7e+142) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	return tmp;
}

Fortran

NOTE: x.im 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 <= 2.35d+126) then
        tmp = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) - (x_46im * ((x_46im * x_46re) + (x_46im * x_46re)))
    else if (x_46im <= 4.7d+142) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (-3.0d0) * (x_46im * (x_46im * x_46re))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.35e+126) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	} else if (x_46_im <= 4.7e+142) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 2.35e+126:
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)))
	elif x_46_im <= 4.7e+142:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.35e+126)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re))));
	elseif (x_46_im <= 4.7e+142)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 2.35e+126)
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	elseif (x_46_im <= 4.7e+142)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 2.35e+126], N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 4.7e+142], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < 2.3499999999999999e126

   1. Initial program 87.4%

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

   if 2.3499999999999999e126 < x.im < 4.7e142

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x.re around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. metadata-eval 0.0%

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

      4. distribute-rgt-out-- 0.0%

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

      5. *-commutative 0.0%

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

      6. fma-def 100.0%

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

      7. distribute-rgt-out-- 100.0%

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

      8. metadata-eval 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*r* 100.0%

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

      11. fma-def 0.0%

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

      12. cube-mult 0.0%

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

      13. unpow2 0.0%

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

      14. distribute-lft-out 100.0%

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

      15. associate-*r* 100.0%

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

      16. unpow2 100.0%

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

      17. *-commutative 100.0%

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

      18. fma-def 100.0%

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

      19. unpow2 100.0%

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

      20. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x.im around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if 4.7e142 < x.im

   1. Initial program 53.8%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in x.re around 0 51.2%

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

      1. *-commutative 51.2%

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

      2. associate-*r* 51.2%

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

      3. *-commutative 51.2%

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

      4. metadata-eval 51.2%

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

      5. distribute-rgt-out-- 51.2%

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

      6. unpow2 51.2%

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

      7. associate-*r* 81.8%

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

      8. distribute-rgt-out-- 81.8%

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

      9. metadata-eval 81.8%

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

      10. *-commutative 81.8%

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

   6. Simplified 81.8%

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

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

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

   8. Taylor expanded in x.re around 0 61.5%

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

      1. *-commutative 61.5%

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

      2. unpow2 61.5%

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

      3. associate-*r* 92.1%

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

      4. *-commutative 92.1%

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

   10. Simplified 92.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.35 \cdot 10^{+126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 4.7 \cdot 10^{+142}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 5: 73.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.6 \cdot 10^{+18} \lor \neg \left(x.re \leq 6.8 \cdot 10^{-69}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -2.6e+18) (not (<= x.re 6.8e-69)))
   (* x.re (* x.re x.re))
   (* -3.0 (* x.re (* x.im x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2.6e+18) || !(x_46_re <= 6.8e-69)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

NOTE: x.im 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 <= (-2.6d+18)) .or. (.not. (x_46re <= 6.8d-69))) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (-3.0d0) * (x_46re * (x_46im * x_46im))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2.6e+18) || !(x_46_re <= 6.8e-69)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -2.6e+18) or not (x_46_re <= 6.8e-69):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -2.6e+18) || !(x_46_re <= 6.8e-69))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -2.6e+18) || ~((x_46_re <= 6.8e-69)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -2.6e+18], N[Not[LessEqual[x$46$re, 6.8e-69]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -2.6e18 or 6.80000000000000016e-69 < x.re

   1. Initial program 77.0%

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

   3. Simplified 74.9%

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

   4. Taylor expanded in x.re around 0 74.9%

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

      1. associate-*r* 74.9%

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

      2. *-commutative 74.9%

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

      3. metadata-eval 74.9%

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

      4. distribute-rgt-out-- 74.9%

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

      5. *-commutative 74.9%

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

      6. fma-def 83.1%

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

      7. distribute-rgt-out-- 83.1%

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

      8. metadata-eval 83.1%

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

      9. unpow2 83.1%

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

      10. associate-*r* 83.1%

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

      11. fma-def 74.9%

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

      12. cube-mult 74.7%

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

      13. unpow2 74.7%

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

      14. distribute-lft-out 85.1%

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

      15. associate-*r* 85.1%

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

      16. unpow2 85.1%

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

      17. *-commutative 85.1%

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

      18. fma-def 85.1%

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

      19. unpow2 85.1%

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

      20. unpow2 85.1%

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

   6. Simplified 85.1%

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

   7. Taylor expanded in x.im around 0 79.9%

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

      1. unpow2 79.9%

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

   9. Simplified 79.9%

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

   if -2.6e18 < x.re < 6.80000000000000016e-69

   1. Initial program 86.8%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x.re around 0 86.8%

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

      1. *-commutative 86.8%

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

      2. associate-*r* 86.8%

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

      3. *-commutative 86.8%

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

      4. metadata-eval 86.8%

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

      5. distribute-rgt-out-- 86.8%

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

      6. unpow2 86.8%

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

      7. associate-*r* 99.7%

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

      8. distribute-rgt-out-- 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x.re around 0 79.7%

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

      1. unpow2 79.7%

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

   9. Simplified 79.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.6 \cdot 10^{+18} \lor \neg \left(x.re \leq 6.8 \cdot 10^{-69}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \]

Alternative 6: 79.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -7 \cdot 10^{+14} \lor \neg \left(x.re \leq 5.1 \cdot 10^{-49}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -7e+14) (not (<= x.re 5.1e-49)))
   (* x.re (* x.re x.re))
   (* -3.0 (* x.im (* x.im x.re)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -7e+14) || !(x_46_re <= 5.1e-49)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	return tmp;
}

Fortran

NOTE: x.im 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 <= (-7d+14)) .or. (.not. (x_46re <= 5.1d-49))) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (-3.0d0) * (x_46im * (x_46im * x_46re))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -7e+14) || !(x_46_re <= 5.1e-49)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -7e+14) or not (x_46_re <= 5.1e-49):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -7e+14) || !(x_46_re <= 5.1e-49))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -7e+14) || ~((x_46_re <= 5.1e-49)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -7e+14], N[Not[LessEqual[x$46$re, 5.1e-49]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -7e14 or 5.10000000000000026e-49 < x.re

   1. Initial program 77.3%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x.re around 0 75.2%

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

      1. associate-*r* 75.2%

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

      2. *-commutative 75.2%

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

      3. metadata-eval 75.2%

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

      4. distribute-rgt-out-- 75.2%

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

      5. *-commutative 75.2%

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

      6. fma-def 83.7%

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

      7. distribute-rgt-out-- 83.7%

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

      8. metadata-eval 83.7%

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

      9. unpow2 83.7%

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

      10. associate-*r* 83.7%

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

      11. fma-def 75.2%

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

      12. cube-mult 75.0%

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

      13. unpow2 75.0%

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

      14. distribute-lft-out 85.9%

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

      15. associate-*r* 85.8%

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

      16. unpow2 85.8%

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

      17. *-commutative 85.8%

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

      18. fma-def 85.9%

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

      19. unpow2 85.9%

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

      20. unpow2 85.9%

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

   6. Simplified 85.9%

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

   7. Taylor expanded in x.im around 0 81.3%

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

      1. unpow2 81.3%

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

   9. Simplified 81.3%

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

   if -7e14 < x.re < 5.10000000000000026e-49

   1. Initial program 86.0%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in x.re around 0 85.9%

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

      1. *-commutative 85.9%

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

      2. associate-*r* 86.0%

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

      3. *-commutative 86.0%

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

      4. metadata-eval 86.0%

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

      5. distribute-rgt-out-- 86.0%

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

      6. unpow2 86.0%

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

      7. associate-*r* 99.7%

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

      8. distribute-rgt-out-- 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x.re around 0 99.7%

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

   8. Taylor expanded in x.re around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. associate-*r* 90.8%

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

      4. *-commutative 90.8%

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

   10. Simplified 90.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -7 \cdot 10^{+14} \lor \neg \left(x.re \leq 5.1 \cdot 10^{-49}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 7: 58.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x.im 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_46re * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

3. Simplified 80.5%

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

4. Taylor expanded in x.re around 0 80.5%

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

   1. associate-*r* 80.5%

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

   2. *-commutative 80.5%

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

   3. metadata-eval 80.5%

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

   4. distribute-rgt-out-- 80.5%

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

   5. *-commutative 80.5%

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

   6. fma-def 84.8%

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

   7. distribute-rgt-out-- 84.8%

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

   8. metadata-eval 84.8%

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

   9. unpow2 84.8%

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

   10. associate-*r* 84.8%

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

   11. fma-def 80.5%

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

   12. cube-mult 80.4%

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

   13. unpow2 80.4%

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

   14. distribute-lft-out 85.9%

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

   15. associate-*r* 85.9%

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

   16. unpow2 85.9%

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

   17. *-commutative 85.9%

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

   18. fma-def 85.9%

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

   19. unpow2 85.9%

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

   20. unpow2 85.9%

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

6. Simplified 85.9%

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

7. Taylor expanded in x.im around 0 57.8%

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

   1. unpow2 57.8%

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

9. Simplified 57.8%

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

10. Final simplification 57.8%

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

Alternative 8: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ -38 \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 -38.0)

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	return -38.0;
}

Fortran

NOTE: x.im 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 = -38.0d0
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	return -38.0;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	return -38.0

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	return -38.0
end

MATLAB

x.im = abs(x.im)
function tmp = code(x_46_re, x_46_im)
	tmp = -38.0;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := -38.0

Derivation

1. Initial program 81.6%

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

3. Simplified 80.5%

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

4. Taylor expanded in x.re around 0 80.5%

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

   1. *-commutative 80.5%

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

   2. associate-*r* 80.5%

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

   3. *-commutative 80.5%

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

   4. metadata-eval 80.5%

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

   5. distribute-rgt-out-- 80.5%

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

   6. unpow2 80.5%

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

   7. associate-*r* 87.4%

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

   8. distribute-rgt-out-- 87.4%

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

   9. metadata-eval 87.4%

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

   10. *-commutative 87.4%

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

6. Simplified 87.4%

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

8. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{-38} \]

9. Final simplification 2.6%

   \[\leadsto -38 \]

Alternative 9: 15.0% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

3. Simplified 80.5%

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

4. Taylor expanded in x.re around 0 80.5%

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

   1. *-commutative 80.5%

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

   2. associate-*r* 80.5%

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

   3. *-commutative 80.5%

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

   4. metadata-eval 80.5%

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

   5. distribute-rgt-out-- 80.5%

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

   6. unpow2 80.5%

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

   7. associate-*r* 87.4%

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

   8. distribute-rgt-out-- 87.4%

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

   9. metadata-eval 87.4%

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

   10. *-commutative 87.4%

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

6. Simplified 87.4%

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

8. Applied egg-rr 13.5%

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

9. Final simplification 13.5%

   \[\leadsto 0 \]

Developer target: 86.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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