math.cube on complex, real part

Percentage Accurate: 82.9% → 98.4%

Time: 5.2s

Alternatives: 9

Speedup: 1.4×

• Unsound rule application detected in e-graph. Results may not be sound.

• 44.0% 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: 82.9% 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: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ t_1 := x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im \cdot x.im\right) \cdot \left(x.re \cdot 2\right)\\ \mathbf{if}\;x.re \leq -2 \cdot 10^{+119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -1.25 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq 2.8 \cdot 10^{-177}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{elif}\;x.re \leq 6 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (+ x.re x.im) (* x.re (- x.re x.im))))
        (t_1
         (-
          (* x.re (- (* x.re x.re) (* x.im x.im)))
          (* (* x.im x.im) (* x.re 2.0)))))
   (if (<= x.re -2e+119)
     t_0
     (if (<= x.re -1.25e-104)
       t_1
       (if (<= x.re 2.8e-177)
         (* x.im (* (* x.re x.im) -3.0))
         (if (<= x.re 6e+57) t_1 t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	double t_1 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - ((x_46_im * x_46_im) * (x_46_re * 2.0));
	double tmp;
	if (x_46_re <= -2e+119) {
		tmp = t_0;
	} else if (x_46_re <= -1.25e-104) {
		tmp = t_1;
	} else if (x_46_re <= 2.8e-177) {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	} else if (x_46_re <= 6e+57) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46re + x_46im) * (x_46re * (x_46re - x_46im))
    t_1 = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) - ((x_46im * x_46im) * (x_46re * 2.0d0))
    if (x_46re <= (-2d+119)) then
        tmp = t_0
    else if (x_46re <= (-1.25d-104)) then
        tmp = t_1
    else if (x_46re <= 2.8d-177) then
        tmp = x_46im * ((x_46re * x_46im) * (-3.0d0))
    else if (x_46re <= 6d+57) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	double t_1 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - ((x_46_im * x_46_im) * (x_46_re * 2.0));
	double tmp;
	if (x_46_re <= -2e+119) {
		tmp = t_0;
	} else if (x_46_re <= -1.25e-104) {
		tmp = t_1;
	} else if (x_46_re <= 2.8e-177) {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	} else if (x_46_re <= 6e+57) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))
	t_1 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - ((x_46_im * x_46_im) * (x_46_re * 2.0))
	tmp = 0
	if x_46_re <= -2e+119:
		tmp = t_0
	elif x_46_re <= -1.25e-104:
		tmp = t_1
	elif x_46_re <= 2.8e-177:
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0)
	elif x_46_re <= 6e+57:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re * Float64(x_46_re - x_46_im)))
	t_1 = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) - Float64(Float64(x_46_im * x_46_im) * Float64(x_46_re * 2.0)))
	tmp = 0.0
	if (x_46_re <= -2e+119)
		tmp = t_0;
	elseif (x_46_re <= -1.25e-104)
		tmp = t_1;
	elseif (x_46_re <= 2.8e-177)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * -3.0));
	elseif (x_46_re <= 6e+57)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	t_1 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - ((x_46_im * x_46_im) * (x_46_re * 2.0));
	tmp = 0.0;
	if (x_46_re <= -2e+119)
		tmp = t_0;
	elseif (x_46_re <= -1.25e-104)
		tmp = t_1;
	elseif (x_46_re <= 2.8e-177)
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	elseif (x_46_re <= 6e+57)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(x$46$re * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -2e+119], t$95$0, If[LessEqual[x$46$re, -1.25e-104], t$95$1, If[LessEqual[x$46$re, 2.8e-177], N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 6e+57], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.99999999999999989e119 or 5.9999999999999999e57 < x.re

   1. Initial program 65.9%

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

      1. *-commutative 65.9%

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

      2. *-commutative 65.9%

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

      3. distribute-rgt-in 65.9%

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

      4. *-commutative 65.9%

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

      5. *-commutative 65.9%

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

   3. Applied egg-rr 65.9%

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

      1. sub-neg 65.9%

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

      2. difference-of-squares 72.3%

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

      3. associate-*l* 72.3%

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

      4. distribute-neg-in 72.3%

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

      5. associate-*l* 72.3%

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

      6. distribute-lft-neg-out 72.3%

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

      7. add-sqr-sqrt 30.9%

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

      8. sqrt-unprod 34.0%

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

      9. sqr-neg 34.0%

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

      10. sqrt-prod 28.7%

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

      11. add-sqr-sqrt 53.2%

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

      12. associate-*l* 53.2%

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

      13. sub-neg 53.2%

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

      14. +-inverses 100.0%

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{0} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + 0} \]
   6. Step-by-step derivation

      1. +-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -1.99999999999999989e119 < x.re < -1.24999999999999995e-104 or 2.79999999999999987e-177 < x.re < 5.9999999999999999e57

   1. Initial program 99.7%

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

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

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

   4. Simplified 99.7%

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

   if -1.24999999999999995e-104 < x.re < 2.79999999999999987e-177

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. *-commutative 85.7%

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

      3. distribute-rgt-in 85.7%

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

      4. *-commutative 85.7%

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

      5. *-commutative 85.7%

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

   3. Applied egg-rr 85.7%

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

   4. Taylor expanded in x.re around 0 85.6%

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

      1. unpow2 85.6%

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

      2. unpow2 85.6%

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

      3. distribute-rgt-out-- 85.6%

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

      4. metadata-eval 85.6%

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

      5. associate-*r* 85.6%

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

      6. associate-*r* 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*l* 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.re \leq -1.25 \cdot 10^{-104}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im \cdot x.im\right) \cdot \left(x.re \cdot 2\right)\\ \mathbf{elif}\;x.re \leq 2.8 \cdot 10^{-177}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{elif}\;x.re \leq 6 \cdot 10^{+57}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im \cdot x.im\right) \cdot \left(x.re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 2: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.im\right)\\ t_1 := x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{if}\;t_1 - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{+240}:\\ \;\;\;\;t_1 - \left(t_0 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (* x.re x.im)))
        (t_1 (* x.re (- (* x.re x.re) (* x.im x.im)))))
   (if (<= (- t_1 (* x.im (+ (* x.re x.im) (* x.re x.im)))) 2e+240)
     (- t_1 (+ t_0 t_0))
     (* (+ x.re x.im) (* x.re (- x.re x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * (x_46_re * x_46_im);
	double t_1 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	double tmp;
	if ((t_1 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= 2e+240) {
		tmp = t_1 - (t_0 + t_0);
	} else {
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_46im * (x_46re * x_46im)
    t_1 = x_46re * ((x_46re * x_46re) - (x_46im * x_46im))
    if ((t_1 - (x_46im * ((x_46re * x_46im) + (x_46re * x_46im)))) <= 2d+240) then
        tmp = t_1 - (t_0 + t_0)
    else
        tmp = (x_46re + x_46im) * (x_46re * (x_46re - x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * (x_46_re * x_46_im);
	double t_1 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	double tmp;
	if ((t_1 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= 2e+240) {
		tmp = t_1 - (t_0 + t_0);
	} else {
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_im * (x_46_re * x_46_im)
	t_1 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))
	tmp = 0
	if (t_1 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= 2e+240:
		tmp = t_1 - (t_0 + t_0)
	else:
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_im * Float64(x_46_re * x_46_im))
	t_1 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (Float64(t_1 - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= 2e+240)
		tmp = Float64(t_1 - Float64(t_0 + t_0));
	else
		tmp = Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re * Float64(x_46_re - x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_im * (x_46_re * x_46_im);
	t_1 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	tmp = 0.0;
	if ((t_1 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= 2e+240)
		tmp = t_1 - (t_0 + t_0);
	else
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+240], N[(t$95$1 - N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 2.00000000000000003e240

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

      2. *-commutative 98.2%

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

      3. distribute-rgt-in 98.2%

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

      4. *-commutative 98.2%

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

      5. *-commutative 98.2%

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

   3. Applied egg-rr 98.2%

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

   if 2.00000000000000003e240 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 49.3%

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

      1. *-commutative 49.3%

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

      2. *-commutative 49.3%

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

      3. distribute-rgt-in 49.3%

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

      4. *-commutative 49.3%

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

      5. *-commutative 49.3%

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

   3. Applied egg-rr 49.3%

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

      1. sub-neg 49.3%

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

      2. difference-of-squares 56.9%

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

      3. associate-*l* 66.6%

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

      4. distribute-neg-in 66.6%

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

      5. associate-*l* 56.9%

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

      6. distribute-lft-neg-out 56.9%

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

      7. add-sqr-sqrt 26.2%

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

      8. sqrt-unprod 46.2%

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

      9. sqr-neg 46.2%

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

      10. sqrt-prod 26.9%

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

      11. add-sqr-sqrt 27.0%

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

      12. associate-*l* 28.7%

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

      13. sub-neg 28.7%

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

      14. +-inverses 91.6%

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{0} \]

   5. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + 0} \]
   6. Step-by-step derivation

      1. +-rgt-identity 91.6%

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

      2. *-commutative 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{+240}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im \cdot \left(x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 3: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re (- (* x.re x.re) (* x.im x.im)))
          (* x.im (+ (* x.re x.im) (* x.re x.im))))))
   (if (<= t_0 2e+240) t_0 (* (+ x.re x.im) (* x.re (- x.re x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double tmp;
	if (t_0 <= 2e+240) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) - (x_46im * ((x_46re * x_46im) + (x_46re * x_46im)))
    if (t_0 <= 2d+240) then
        tmp = t_0
    else
        tmp = (x_46re + x_46im) * (x_46re * (x_46re - x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double tmp;
	if (t_0 <= 2e+240) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	tmp = 0
	if t_0 <= 2e+240:
		tmp = t_0
	else:
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = 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_re * x_46_im) + Float64(x_46_re * x_46_im))))
	tmp = 0.0
	if (t_0 <= 2e+240)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re * Float64(x_46_re - x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	tmp = 0.0;
	if (t_0 <= 2e+240)
		tmp = t_0;
	else
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = 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$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+240], t$95$0, N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 2.00000000000000003e240

   1. Initial program 98.2%

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

   if 2.00000000000000003e240 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 49.3%

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

      1. *-commutative 49.3%

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

      2. *-commutative 49.3%

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

      3. distribute-rgt-in 49.3%

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

      4. *-commutative 49.3%

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

      5. *-commutative 49.3%

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

   3. Applied egg-rr 49.3%

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

      1. sub-neg 49.3%

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

      2. difference-of-squares 56.9%

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

      3. associate-*l* 66.6%

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

      4. distribute-neg-in 66.6%

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

      5. associate-*l* 56.9%

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

      6. distribute-lft-neg-out 56.9%

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

      7. add-sqr-sqrt 26.2%

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

      8. sqrt-unprod 46.2%

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

      9. sqr-neg 46.2%

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

      10. sqrt-prod 26.9%

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

      11. add-sqr-sqrt 27.0%

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

      12. associate-*l* 28.7%

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

      13. sub-neg 28.7%

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

      14. +-inverses 91.6%

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{0} \]

   5. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + 0} \]
   6. Step-by-step derivation

      1. +-rgt-identity 91.6%

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

      2. *-commutative 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{+240}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 4: 90.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -6.6 \cdot 10^{-88} \lor \neg \left(x.re \leq 2.7 \cdot 10^{+36}\right):\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -6.6e-88) (not (<= x.re 2.7e+36)))
   (* (+ x.re x.im) (* x.re (- x.re x.im)))
   (* x.im (* (* x.re x.im) -3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -6.6e-88) || !(x_46_re <= 2.7e+36)) {
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-6.6d-88)) .or. (.not. (x_46re <= 2.7d+36))) then
        tmp = (x_46re + x_46im) * (x_46re * (x_46re - x_46im))
    else
        tmp = x_46im * ((x_46re * x_46im) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -6.6e-88) || !(x_46_re <= 2.7e+36)) {
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -6.6e-88) or not (x_46_re <= 2.7e+36):
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))
	else:
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -6.6e-88) || !(x_46_re <= 2.7e+36))
		tmp = Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re * Float64(x_46_re - x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -6.6e-88) || ~((x_46_re <= 2.7e+36)))
		tmp = (x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im));
	else
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -6.6e-88], N[Not[LessEqual[x$46$re, 2.7e+36]], $MachinePrecision]], N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -6.59999999999999987e-88 or 2.7000000000000001e36 < x.re

   1. Initial program 77.5%

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

      1. *-commutative 77.5%

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

      2. *-commutative 77.5%

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

      3. distribute-rgt-in 77.5%

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

      4. *-commutative 77.5%

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

      5. *-commutative 77.5%

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

   3. Applied egg-rr 77.5%

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

      1. sub-neg 77.5%

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

      2. difference-of-squares 81.7%

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

      3. associate-*l* 81.7%

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

      4. distribute-neg-in 81.7%

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

      5. associate-*l* 81.7%

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

      6. distribute-lft-neg-out 81.7%

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

      7. add-sqr-sqrt 52.3%

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

      8. sqrt-unprod 55.2%

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

      9. sqr-neg 55.2%

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

      10. sqrt-prod 19.7%

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

      11. add-sqr-sqrt 57.4%

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

      12. associate-*l* 57.4%

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

      13. sub-neg 57.4%

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

      14. +-inverses 95.9%

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{0} \]

   5. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + 0} \]
   6. Step-by-step derivation

      1. +-rgt-identity 95.9%

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

      2. *-commutative 95.9%

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

   7. Simplified 95.9%

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

   if -6.59999999999999987e-88 < x.re < 2.7000000000000001e36

   1. Initial program 90.6%

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

      1. *-commutative 90.6%

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

      2. *-commutative 90.6%

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

      3. distribute-rgt-in 90.6%

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

      4. *-commutative 90.6%

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

      5. *-commutative 90.6%

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

   3. Applied egg-rr 90.6%

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

   4. Taylor expanded in x.re around 0 81.1%

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

      1. unpow2 81.1%

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

      2. unpow2 81.1%

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

      3. distribute-rgt-out-- 81.1%

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

      4. metadata-eval 81.1%

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

      5. associate-*r* 81.1%

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

      6. associate-*r* 90.3%

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

      7. *-commutative 90.3%

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

      8. associate-*l* 90.3%

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

   6. Simplified 90.3%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -6.6 \cdot 10^{-88} \lor \neg \left(x.re \leq 2.7 \cdot 10^{+36}\right):\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \]

Alternative 5: 60.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{+154} \lor \neg \left(x.re \leq 4.6 \cdot 10^{+149}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -5e+154) (not (<= x.re 4.6e+149)))
   (* x.im (* x.re x.im))
   (* x.im (* x.re (* x.im -3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -5e+154) || !(x_46_re <= 4.6e+149)) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-5d+154)) .or. (.not. (x_46re <= 4.6d+149))) then
        tmp = x_46im * (x_46re * x_46im)
    else
        tmp = x_46im * (x_46re * (x_46im * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -5e+154) || !(x_46_re <= 4.6e+149)) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -5e+154) or not (x_46_re <= 4.6e+149):
		tmp = x_46_im * (x_46_re * x_46_im)
	else:
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -5e+154) || !(x_46_re <= 4.6e+149))
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_im));
	else
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -5e+154) || ~((x_46_re <= 4.6e+149)))
		tmp = x_46_im * (x_46_re * x_46_im);
	else
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -5e+154], N[Not[LessEqual[x$46$re, 4.6e+149]], $MachinePrecision]], N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -5.00000000000000004e154 or 4.5999999999999997e149 < x.re

   1. Initial program 50.0%

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

   2. Taylor expanded in x.re around 0 10.1%

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

   4. Simplified 10.1%

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

      1. *-commutative 10.1%

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

      2. *-commutative 10.1%

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

      3. distribute-rgt-out 10.1%

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

      4. associate--r+ 10.1%

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

      5. add-sqr-sqrt 5.1%

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

      6. sqrt-unprod 14.5%

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

      7. sqr-neg 14.5%

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

      8. sqrt-prod 0.1%

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

      9. add-sqr-sqrt 0.4%

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

      10. associate-*l* 0.4%

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

      11. +-inverses 10.1%

          \[\leadsto \color{blue}{0} - \left(x.re \cdot x.im\right) \cdot x.im \]

      12. neg-sub0 10.1%

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

      13. associate-*l* 10.1%

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

      14. distribute-lft-neg-out 10.1%

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

      15. add-sqr-sqrt 5.1%

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

      16. sqrt-unprod 29.1%

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

      17. sqr-neg 29.1%

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

      18. sqrt-prod 15.3%

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

      19. add-sqr-sqrt 42.1%

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

      20. associate-*l* 42.1%

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

   6. Applied egg-rr 42.1%

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

   if -5.00000000000000004e154 < x.re < 4.5999999999999997e149

   1. Initial program 93.9%

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

      1. *-commutative 93.9%

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

      2. *-commutative 93.9%

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

      3. distribute-rgt-in 93.9%

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

      4. *-commutative 93.9%

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

      5. *-commutative 93.9%

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

   3. Applied egg-rr 93.9%

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

   4. Taylor expanded in x.re around 0 62.7%

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

      1. unpow2 62.7%

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

      2. unpow2 62.7%

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

      3. distribute-rgt-out-- 62.7%

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

      4. metadata-eval 62.7%

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

      5. associate-*r* 62.7%

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

      6. associate-*r* 68.1%

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

      7. *-commutative 68.1%

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

      8. associate-*l* 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in x.re around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. associate-*r* 68.1%

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

   9. Simplified 68.1%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{+154} \lor \neg \left(x.re \leq 4.6 \cdot 10^{+149}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 6: 38.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -9.2 \cdot 10^{+153} \lor \neg \left(x.re \leq 6.1 \cdot 10^{+149}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(-x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -9.2e+153) (not (<= x.re 6.1e+149)))
   (* x.im (* x.re x.im))
   (* x.re (- (* x.im x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -9.2e+153) || !(x_46_re <= 6.1e+149)) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = x_46_re * -(x_46_im * x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-9.2d+153)) .or. (.not. (x_46re <= 6.1d+149))) then
        tmp = x_46im * (x_46re * x_46im)
    else
        tmp = x_46re * -(x_46im * x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -9.2e+153) || !(x_46_re <= 6.1e+149)) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = x_46_re * -(x_46_im * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -9.2e+153) or not (x_46_re <= 6.1e+149):
		tmp = x_46_im * (x_46_re * x_46_im)
	else:
		tmp = x_46_re * -(x_46_im * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -9.2e+153) || !(x_46_re <= 6.1e+149))
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_im));
	else
		tmp = Float64(x_46_re * Float64(-Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -9.2e+153) || ~((x_46_re <= 6.1e+149)))
		tmp = x_46_im * (x_46_re * x_46_im);
	else
		tmp = x_46_re * -(x_46_im * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -9.2e+153], N[Not[LessEqual[x$46$re, 6.1e+149]], $MachinePrecision]], N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re * (-N[(x$46$im * x$46$im), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -9.2000000000000005e153 or 6.0999999999999999e149 < x.re

   1. Initial program 50.0%

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

   2. Taylor expanded in x.re around 0 10.1%

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

   4. Simplified 10.1%

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

      1. *-commutative 10.1%

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

      2. *-commutative 10.1%

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

      3. distribute-rgt-out 10.1%

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

      4. associate--r+ 10.1%

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

      5. add-sqr-sqrt 5.1%

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

      6. sqrt-unprod 14.5%

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

      7. sqr-neg 14.5%

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

      8. sqrt-prod 0.1%

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

      9. add-sqr-sqrt 0.4%

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

      10. associate-*l* 0.4%

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

      11. +-inverses 10.1%

          \[\leadsto \color{blue}{0} - \left(x.re \cdot x.im\right) \cdot x.im \]

      12. neg-sub0 10.1%

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

      13. associate-*l* 10.1%

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

      14. distribute-lft-neg-out 10.1%

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

      15. add-sqr-sqrt 5.1%

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

      16. sqrt-unprod 29.1%

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

      17. sqr-neg 29.1%

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

      18. sqrt-prod 15.3%

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

      19. add-sqr-sqrt 42.1%

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

      20. associate-*l* 42.1%

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

   6. Applied egg-rr 42.1%

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

   if -9.2000000000000005e153 < x.re < 6.0999999999999999e149

   1. Initial program 93.9%

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

   2. Taylor expanded in x.re around 0 62.8%

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

   4. Simplified 62.8%

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

      1. *-commutative 62.8%

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

      2. *-commutative 62.8%

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

      3. distribute-rgt-out 62.8%

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

      4. associate--r+ 62.8%

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

      5. add-sqr-sqrt 33.8%

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

      6. sqrt-unprod 37.2%

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

      7. sqr-neg 37.2%

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

      8. sqrt-prod 9.4%

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

      9. add-sqr-sqrt 22.4%

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

      10. associate-*l* 23.4%

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

      11. +-inverses 40.4%

          \[\leadsto \color{blue}{0} - \left(x.re \cdot x.im\right) \cdot x.im \]

      12. neg-sub0 40.4%

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

      13. associate-*l* 39.7%

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

      14. distribute-rgt-neg-in 39.7%

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

   6. Applied egg-rr 39.7%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -9.2 \cdot 10^{+153} \lor \neg \left(x.re \leq 6.1 \cdot 10^{+149}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(-x.im \cdot x.im\right)\\ \end{array} \]

Alternative 7: 39.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.45 \cdot 10^{+154} \lor \neg \left(x.re \leq 5.2 \cdot 10^{+149}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -1.45e+154) (not (<= x.re 5.2e+149)))
   (* x.im (* x.re x.im))
   (* x.im (* x.re (- x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.45e+154) || !(x_46_re <= 5.2e+149)) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * -x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-1.45d+154)) .or. (.not. (x_46re <= 5.2d+149))) then
        tmp = x_46im * (x_46re * x_46im)
    else
        tmp = x_46im * (x_46re * -x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.45e+154) || !(x_46_re <= 5.2e+149)) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * -x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -1.45e+154) or not (x_46_re <= 5.2e+149):
		tmp = x_46_im * (x_46_re * x_46_im)
	else:
		tmp = x_46_im * (x_46_re * -x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -1.45e+154) || !(x_46_re <= 5.2e+149))
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_im));
	else
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -1.45e+154) || ~((x_46_re <= 5.2e+149)))
		tmp = x_46_im * (x_46_re * x_46_im);
	else
		tmp = x_46_im * (x_46_re * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -1.45e+154], N[Not[LessEqual[x$46$re, 5.2e+149]], $MachinePrecision]], N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.4499999999999999e154 or 5.19999999999999957e149 < x.re

   1. Initial program 50.0%

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

   2. Taylor expanded in x.re around 0 10.1%

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

   4. Simplified 10.1%

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

      1. *-commutative 10.1%

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

      2. *-commutative 10.1%

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

      3. distribute-rgt-out 10.1%

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

      4. associate--r+ 10.1%

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

      5. add-sqr-sqrt 5.1%

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

      6. sqrt-unprod 14.5%

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

      7. sqr-neg 14.5%

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

      8. sqrt-prod 0.1%

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

      9. add-sqr-sqrt 0.4%

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

      10. associate-*l* 0.4%

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

      11. +-inverses 10.1%

          \[\leadsto \color{blue}{0} - \left(x.re \cdot x.im\right) \cdot x.im \]

      12. neg-sub0 10.1%

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

      13. associate-*l* 10.1%

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

      14. distribute-lft-neg-out 10.1%

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

      15. add-sqr-sqrt 5.1%

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

      16. sqrt-unprod 29.1%

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

      17. sqr-neg 29.1%

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

      18. sqrt-prod 15.3%

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

      19. add-sqr-sqrt 42.1%

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

      20. associate-*l* 42.1%

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

   6. Applied egg-rr 42.1%

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

   if -1.4499999999999999e154 < x.re < 5.19999999999999957e149

   1. Initial program 93.9%

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

   2. Taylor expanded in x.re around 0 62.8%

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

   4. Simplified 62.8%

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

      1. *-commutative 62.8%

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

      2. *-commutative 62.8%

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

      3. distribute-rgt-out 62.8%

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

      4. associate--r+ 62.8%

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

      5. add-sqr-sqrt 33.8%

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

      6. sqrt-unprod 37.2%

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

      7. sqr-neg 37.2%

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

      8. sqrt-prod 9.4%

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

      9. add-sqr-sqrt 22.4%

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

      10. associate-*l* 23.4%

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

      11. +-inverses 40.4%

          \[\leadsto \color{blue}{0} - \left(x.re \cdot x.im\right) \cdot x.im \]

      12. neg-sub0 40.4%

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

      13. distribute-rgt-neg-in 40.4%

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

   6. Applied egg-rr 40.4%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.45 \cdot 10^{+154} \lor \neg \left(x.re \leq 5.2 \cdot 10^{+149}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 8: 23.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

2. Taylor expanded in x.re around 0 50.0%

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

4. Simplified 50.0%

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

   1. *-commutative 50.0%

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

   2. *-commutative 50.0%

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

   3. distribute-rgt-out 50.0%

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

   4. associate--r+ 50.0%

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

   5. add-sqr-sqrt 26.8%

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

   6. sqrt-unprod 31.7%

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

   7. sqr-neg 31.7%

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

   8. sqrt-prod 7.1%

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

   9. add-sqr-sqrt 17.1%

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

   10. associate-*l* 17.8%

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

   11. +-inverses 33.0%

       \[\leadsto \color{blue}{0} - \left(x.re \cdot x.im\right) \cdot x.im \]

   12. neg-sub0 33.0%

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

   13. associate-*l* 32.5%

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

   14. distribute-lft-neg-out 32.5%

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

   15. *-commutative 32.5%

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

   16. add-sqr-sqrt 17.9%

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

   17. sqrt-unprod 28.1%

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

   18. sqr-neg 28.1%

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

   19. sqrt-prod 9.2%

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

   20. add-sqr-sqrt 24.5%

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

6. Applied egg-rr 24.5%

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

7. Final simplification 24.5%

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

Alternative 9: 23.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return 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_46im * (x_46re * x_46im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

2. Taylor expanded in x.re around 0 50.0%

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

4. Simplified 50.0%

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

   1. *-commutative 50.0%

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

   2. *-commutative 50.0%

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

   3. distribute-rgt-out 50.0%

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

   4. associate--r+ 50.0%

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

   5. add-sqr-sqrt 26.8%

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

   6. sqrt-unprod 31.7%

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

   7. sqr-neg 31.7%

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

   8. sqrt-prod 7.1%

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

   9. add-sqr-sqrt 17.1%

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

   10. associate-*l* 17.8%

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

   11. +-inverses 33.0%

       \[\leadsto \color{blue}{0} - \left(x.re \cdot x.im\right) \cdot x.im \]

   12. neg-sub0 33.0%

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

   13. associate-*l* 32.5%

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

   14. distribute-lft-neg-out 32.5%

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

   15. add-sqr-sqrt 17.9%

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

   16. sqrt-unprod 28.1%

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

   17. sqr-neg 28.1%

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

   18. sqrt-prod 9.2%

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

   19. add-sqr-sqrt 24.5%

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

   20. associate-*l* 24.6%

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

6. Applied egg-rr 24.6%

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

7. Final simplification 24.6%

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

Developer target: 87.4% 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 2023174 
(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)))