math.cube on complex, real part

Percentage Accurate: 82.6% → 98.2%

Time: 8.8s

Alternatives: 6

Speedup: 1.7×

• 44.6% 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: 82.6% 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: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 9 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) + x.im \cdot \left(x.im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.re + x.im \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 9e+148)
   (* x.re (+ (* (- x.re x.im) (+ x.im x.re)) (* x.im (* x.im -2.0))))
   (* (* x.im x.re) (+ x.re (* x.im -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 9e+148) {
		tmp = x_46_re * (((x_46_re - x_46_im) * (x_46_im + x_46_re)) + (x_46_im * (x_46_im * -2.0)));
	} else {
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0));
	}
	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) :: tmp
    if (x_46im <= 9d+148) then
        tmp = x_46re * (((x_46re - x_46im) * (x_46im + x_46re)) + (x_46im * (x_46im * (-2.0d0))))
    else
        tmp = (x_46im * x_46re) * (x_46re + (x_46im * (-3.0d0)))
    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 <= 9e+148) {
		tmp = x_46_re * (((x_46_re - x_46_im) * (x_46_im + x_46_re)) + (x_46_im * (x_46_im * -2.0)));
	} else {
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 9e+148:
		tmp = x_46_re * (((x_46_re - x_46_im) * (x_46_im + x_46_re)) + (x_46_im * (x_46_im * -2.0)))
	else:
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 9e+148)
		tmp = Float64(x_46_re * Float64(Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re)) + Float64(x_46_im * Float64(x_46_im * -2.0))));
	else
		tmp = Float64(Float64(x_46_im * x_46_re) * Float64(x_46_re + Float64(x_46_im * -3.0)));
	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 <= 9e+148)
		tmp = x_46_re * (((x_46_re - x_46_im) * (x_46_im + x_46_re)) + (x_46_im * (x_46_im * -2.0)));
	else
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0));
	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, 9e+148], N[(x$46$re * N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 8.99999999999999987e148

   1. Initial program 86.7%

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

      1. sqr-neg 86.7%

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

      2. difference-of-squares 87.2%

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

      3. sub-neg 87.2%

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

      4. associate-*l* 91.6%

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

      5. sub-neg 91.6%

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

      6. remove-double-neg 91.6%

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

      7. +-commutative 91.6%

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

      8. *-commutative 91.6%

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

      9. *-commutative 91.6%

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

      10. distribute-rgt-out 91.6%

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

   3. Simplified 91.6%

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

      1. add-cube-cbrt 91.1%

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

      2. pow3 91.2%

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

      3. *-commutative 91.2%

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

   5. Applied egg-rr 91.2%

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

      1. cancel-sign-sub-inv 91.2%

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

      2. unpow3 91.1%

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

      3. add-cube-cbrt 91.6%

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

      4. *-commutative 91.6%

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

      5. associate-*l* 87.2%

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

      6. difference-of-squares 86.7%

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

      7. associate-*r* 86.7%

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

   7. Applied egg-rr 86.7%

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

   9. Simplified 93.6%

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

   if 8.99999999999999987e148 < x.im

   1. Initial program 38.3%

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

      1. sqr-neg 38.3%

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

      2. difference-of-squares 54.3%

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

      3. sub-neg 54.3%

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

      4. associate-*l* 91.8%

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

      5. sub-neg 91.8%

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

      6. remove-double-neg 91.8%

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

      7. +-commutative 91.8%

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

      8. *-commutative 91.8%

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

      9. *-commutative 91.8%

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

      10. distribute-rgt-out 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in x.re around 0 91.8%

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

      1. *-commutative 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in x.re around 0 46.3%

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

      1. *-commutative 46.3%

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

      2. unpow2 46.3%

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

      3. associate-*r* 46.3%

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

      4. *-commutative 46.3%

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

      5. distribute-rgt-out-- 46.3%

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

      6. unpow2 46.3%

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

      7. metadata-eval 46.3%

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

      8. associate-*r* 46.3%

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

      9. associate-*r* 83.8%

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

      10. distribute-lft-out 95.8%

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

   9. Simplified 95.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 9 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) + x.im \cdot \left(x.im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.re + x.im \cdot -3\right)\\ \end{array} \]

Alternative 2: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 8.6 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.re + x.im \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 8.6e+148)
   (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))
   (* (* x.im x.re) (+ x.re (* x.im -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 8.6e+148) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0));
	}
	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) :: tmp
    if (x_46im <= 8.6d+148) then
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = (x_46im * x_46re) * (x_46re + (x_46im * (-3.0d0)))
    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 <= 8.6e+148) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 8.6e+148:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	else:
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 8.6e+148)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(Float64(x_46_im * x_46_re) * Float64(x_46_re + Float64(x_46_im * -3.0)));
	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 <= 8.6e+148)
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	else
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0));
	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, 8.6e+148], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 8.6000000000000003e148

   1. Initial program 86.7%

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

   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. Step-by-step derivation

      1. unpow3 85.8%

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

      2. associate-*r* 85.8%

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

      3. associate-*l* 85.9%

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

      4. fma-def 86.7%

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

      5. associate-*l* 86.7%

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

      6. associate-*r* 86.7%

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

      7. associate-*r* 91.1%

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

   5. Applied egg-rr 91.1%

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

      1. fma-udef 90.2%

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

      2. associate-*l* 85.8%

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

      3. associate-*l* 85.8%

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

      4. *-commutative 85.8%

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

      5. distribute-rgt-out 93.2%

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

      6. associate-*l* 93.2%

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

   7. Applied egg-rr 93.2%

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

   if 8.6000000000000003e148 < x.im

   1. Initial program 38.3%

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

      1. sqr-neg 38.3%

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

      2. difference-of-squares 54.3%

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

      3. sub-neg 54.3%

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

      4. associate-*l* 91.8%

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

      5. sub-neg 91.8%

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

      6. remove-double-neg 91.8%

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

      7. +-commutative 91.8%

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

      8. *-commutative 91.8%

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

      9. *-commutative 91.8%

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

      10. distribute-rgt-out 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in x.re around 0 91.8%

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

      1. *-commutative 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in x.re around 0 46.3%

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

      1. *-commutative 46.3%

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

      2. unpow2 46.3%

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

      3. associate-*r* 46.3%

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

      4. *-commutative 46.3%

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

      5. distribute-rgt-out-- 46.3%

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

      6. unpow2 46.3%

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

      7. metadata-eval 46.3%

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

      8. associate-*r* 46.3%

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

      9. associate-*r* 83.8%

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

      10. distribute-lft-out 95.8%

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

   9. Simplified 95.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 8.6 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.re + x.im \cdot -3\right)\\ \end{array} \]

Alternative 3: 78.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -8.6 \cdot 10^{+42} \lor \neg \left(x.re \leq 10^{-28}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -8.6e+42) (not (<= x.re 1e-28)))
   (* 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 <= -8.6e+42) || !(x_46_re <= 1e-28)) {
		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 <= (-8.6d+42)) .or. (.not. (x_46re <= 1d-28))) 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 <= -8.6e+42) || !(x_46_re <= 1e-28)) {
		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 <= -8.6e+42) or not (x_46_re <= 1e-28):
		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 <= -8.6e+42) || !(x_46_re <= 1e-28))
		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 <= -8.6e+42) || ~((x_46_re <= 1e-28)))
		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, -8.6e+42], N[Not[LessEqual[x$46$re, 1e-28]], $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 < -8.5999999999999996e42 or 9.99999999999999971e-29 < x.re

   1. Initial program 77.3%

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

      1. sqr-neg 77.3%

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

      2. difference-of-squares 81.7%

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

      3. sub-neg 81.7%

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

      4. associate-*l* 81.7%

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

      5. sub-neg 81.7%

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

      6. remove-double-neg 81.7%

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

      7. +-commutative 81.7%

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

      8. *-commutative 81.7%

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

      9. *-commutative 81.7%

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

      10. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

      1. add-cube-cbrt 81.4%

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

      2. pow3 81.4%

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

      3. *-commutative 81.4%

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

   5. Applied egg-rr 81.4%

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

      1. cancel-sign-sub-inv 81.4%

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

      2. unpow3 81.4%

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

      3. add-cube-cbrt 81.7%

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

      4. *-commutative 81.7%

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

      5. associate-*l* 81.7%

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

      6. difference-of-squares 77.3%

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

      7. associate-*r* 77.3%

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

   7. Applied egg-rr 77.3%

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

   9. Simplified 94.7%

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

   10. Taylor expanded in x.re around inf 84.3%

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

   12. Simplified 84.3%

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

   if -8.5999999999999996e42 < x.re < 9.99999999999999971e-29

   1. Initial program 85.8%

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

   3. Simplified 85.8%

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

      1. unpow3 85.8%

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

      2. associate-*r* 85.8%

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

      3. associate-*l* 85.8%

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

      4. fma-def 85.8%

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

      5. associate-*l* 85.8%

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

      6. associate-*r* 85.8%

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

      7. associate-*r* 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. fma-udef 99.6%

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

      2. associate-*l* 85.8%

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

      3. associate-*l* 85.8%

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

      4. *-commutative 85.8%

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

      5. distribute-rgt-out 85.8%

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

      6. associate-*l* 85.8%

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in x.re around 0 73.7%

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

   10. Simplified 87.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -8.6 \cdot 10^{+42} \lor \neg \left(x.re \leq 10^{-28}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 4: 88.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.re + x.im \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 2.95e-15)
   (* x.re (* x.re x.re))
   (* (* x.im x.re) (+ x.re (* x.im -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.95e-15) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0));
	}
	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) :: tmp
    if (x_46im <= 2.95d-15) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (x_46im * x_46re) * (x_46re + (x_46im * (-3.0d0)))
    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 <= 2.95e-15) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 2.95e-15:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.95e-15)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(Float64(x_46_im * x_46_re) * Float64(x_46_re + Float64(x_46_im * -3.0)));
	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 <= 2.95e-15)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = (x_46_im * x_46_re) * (x_46_re + (x_46_im * -3.0));
	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, 2.95e-15], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.94999999999999982e-15

   1. Initial program 89.1%

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

      1. sqr-neg 89.1%

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

      2. difference-of-squares 89.6%

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

      3. sub-neg 89.6%

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

      4. associate-*l* 94.7%

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

      5. sub-neg 94.7%

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

      6. remove-double-neg 94.7%

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

      7. +-commutative 94.7%

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

      8. *-commutative 94.7%

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

      9. *-commutative 94.7%

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

      10. distribute-rgt-out 94.7%

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

   3. Simplified 94.7%

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

      1. add-cube-cbrt 94.3%

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

      2. pow3 94.3%

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

      3. *-commutative 94.3%

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

   5. Applied egg-rr 94.3%

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

      1. cancel-sign-sub-inv 94.3%

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

      2. unpow3 94.3%

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

      3. add-cube-cbrt 94.7%

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

      4. *-commutative 94.7%

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

      5. associate-*l* 89.6%

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

      6. difference-of-squares 89.1%

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

      7. associate-*r* 89.1%

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

   7. Applied egg-rr 89.1%

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

   9. Simplified 92.6%

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

   10. Taylor expanded in x.re around inf 68.1%

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

   12. Simplified 68.1%

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

   if 2.94999999999999982e-15 < x.im

   1. Initial program 58.4%

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

      1. sqr-neg 58.4%

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

      2. difference-of-squares 65.2%

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

      3. sub-neg 65.2%

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

      4. associate-*l* 81.0%

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

      5. sub-neg 81.0%

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

      6. remove-double-neg 81.0%

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

      7. +-commutative 81.0%

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

      8. *-commutative 81.0%

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

      9. *-commutative 81.0%

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

      10. distribute-rgt-out 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in x.re around 0 72.6%

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

      1. *-commutative 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in x.re around 0 53.4%

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

      1. *-commutative 53.4%

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

      2. unpow2 53.4%

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

      3. associate-*r* 53.4%

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

      4. *-commutative 53.4%

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

      5. distribute-rgt-out-- 53.4%

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

      6. unpow2 53.4%

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

      7. metadata-eval 53.4%

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

      8. associate-*r* 53.4%

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

      9. associate-*r* 69.2%

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

      10. distribute-lft-out 81.1%

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

   9. Simplified 81.1%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.re + x.im \cdot -3\right)\\ \end{array} \]

Alternative 5: 27.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_im * 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_46im * 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_im * x_46_re);
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	return x_46_re * (x_46_im * 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_im * 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_im * 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$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

   1. sqr-neg 82.0%

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

   2. difference-of-squares 84.0%

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

   3. sub-neg 84.0%

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

   4. associate-*l* 91.6%

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

   5. sub-neg 91.6%

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

   6. remove-double-neg 91.6%

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

   7. +-commutative 91.6%

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

   8. *-commutative 91.6%

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

   9. *-commutative 91.6%

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

   10. distribute-rgt-out 91.6%

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

3. Simplified 91.6%

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

4. Taylor expanded in x.re around 0 60.5%

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

   1. *-commutative 60.5%

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

6. Simplified 60.5%

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

7. Taylor expanded in x.re around inf 27.3%

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

   1. *-commutative 27.3%

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

   2. unpow2 27.3%

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

   3. associate-*r* 25.9%

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

9. Simplified 25.9%

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

10. Final simplification 25.9%

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

Alternative 6: 58.9% 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 82.0%

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

   1. sqr-neg 82.0%

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

   2. difference-of-squares 84.0%

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

   3. sub-neg 84.0%

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

   4. associate-*l* 91.6%

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

   5. sub-neg 91.6%

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

   6. remove-double-neg 91.6%

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

   7. +-commutative 91.6%

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

   8. *-commutative 91.6%

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

   9. *-commutative 91.6%

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

   10. distribute-rgt-out 91.6%

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

3. Simplified 91.6%

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

   1. add-cube-cbrt 91.2%

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

   2. pow3 91.2%

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

   3. *-commutative 91.2%

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

5. Applied egg-rr 91.2%

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

   1. cancel-sign-sub-inv 91.2%

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

   2. unpow3 91.2%

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

   3. add-cube-cbrt 91.6%

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

   4. *-commutative 91.6%

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

   5. associate-*l* 84.0%

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

   6. difference-of-squares 82.0%

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

   7. associate-*r* 82.0%

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

7. Applied egg-rr 82.0%

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

9. Simplified 89.8%

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

10. Taylor expanded in x.re around inf 58.9%

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

12. Simplified 58.9%

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

13. Final simplification 58.9%

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

Developer target: 87.1% 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 2023275 
(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)))